![Operator Theory:
Advances and Applications, Vol. 250, 1–16
Polynomial Internal and External Stability
of Well-posed Linear Systems
El Mustapha Ait Benhassi, Said Boulite, Lahcen Maniar
and Roland Schnaubelt
Abstract. We introduce polynomial stabilizability and detectability of well-
posed systems in the sense that a feedback produces a polynomially stable
C0-semigroup. Using these concepts, the polynomial stability of the given
C0-semigroup governing the state equation can be characterized via polyno-
mial bounds on the transfer function. We further give sufficient conditions for
polynomial stabilizability and detectability in terms of decompositions into
a polynomial stable and an observable part. Our approach relies on a recent
characterization of polynomially stable C0-semigroups on a Hilbert space by
resolvent estimates.
Mathematics Subject Classification (2010). Primary: 93D25. Secondary: 47A55,
47D06, 93C25, 93D15.
Keywords. Internal and external stability, polynomial stability, transfer func-
tion, stabilizability, detectability, well-posed systems.
1. Introduction
Weakly damped or weakly coupled linear wave type equations often have polyno-
mially decaying classical solutions without being exponentially stable, see, e.g., [1],
[2], [4], [5], [8], [15], [16], [17], [18], [23], and the references therein. In these con-
tributions various methods have been used, partly based on resolvent estimates.
Recently this spectral theory has been completed for the case of bounded semi-
groups T (·) in a Hilbert space with generator A. Here one can now characterize the
‘polynomial stability’ T (t)(I − A)−1
≤ ct−1/α
, t ≥ 1, of T (·) by the polynomial
bound R(iτ, A) ≤ c|τ|α
, |τ| ≥ 1, on the resolvent of A. These results are due to
Borichev and Tomilov in [7] and to Batty and Duyckaerts in [6], see also [5], [15]
and [17] for earlier contributions. We describe this theory in the next section. In a
This work is part of a cooperation project supported by DFG (Germany) and CNRST (Morocco).
c
Springer International Publishing Switzerland 2015](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-15-320.jpg)
![2 E.M. Ait Benhassi, S. Boulite, L. Maniar and R. Schnaubelt
polynomial stable system the spectrum of the generator may approach the imagi-
nary axis as Im λ → ±∞. This already indicates that this concept is more subtle
than exponential stability. For instance, so far robustness results for polynomial
stability are restricted to small regularizing perturbations, see [19].
At least for bounded semigroups in a Hilbert space one has now a solid
background which can be used in other areas such as control theory. In the context
of observability this was already done in [11] (based on [5] at that time). In this
paper we start an investigation of polynomial stabilizability and detectability.
Stabilizability is one of the basic concepts and topics of linear systems theory.
Let the state system be governed by a generator A on the state Hilbert space X,
and let Y and U be the observation and the control Hilbert spaces, respectively.
For a moment, we simply consider bounded control and observation operators and
feedbacks. For a bounded control operator B : U −→ X we obtain the system
x
(t) = Ax(t) + Bu(t), t ≥ 0, x(0) = x0, (1.1)
with the control u ∈ L2
loc(R+, U), the initial state x0 ∈ X and the state x(t) ∈ X
at time t ≥ 0. This system is exponentially stabilizable if one can find a (bounded)
feedback F : X −→ U such that the C0-semigroup TBF (·) solving the closed-loop
system
x
(t) = Ax(t) + BFx(t), t ≥ 0, x(0) = x0, (1.2)
is exponentially stable. Observe that A + BF generates TBF (·).
For the dual concept of exponential detectability, one starts with a generator
A and a bounded observation operator C : X −→ Y . The output of this system
is y = CT (·)x0. One then looks for a (bounded) feedback H : Y −→ X such that
the C0-semigroup THC(·) generated by A + HC becomes exponentially stable.
In our paper we allow for unbounded observation operators C defined on
D(A) and control operators B mapping into the larger space X−1 = D(A∗
)∗
,
where the domains are equipped with the respective graph norm. Here one has to
assume that the output map x0 → y and the input map u → x(t) are continuous.
Such systems are called admissible, see the next section for a precise definition
and further information. The monograph [24] investigates these notions in detail.
In this framework one can in particular treat boundary control and observation of
partial differential equations.
In order to use the full system (A, B, C), one also has to assume the bound-
edness of the input-output map u → y. This leads to the concept of a well-posed
system, which was introduced by G. Weiss and others, see Section 2, the recent
survey [25], and, e.g., [22], [27], [28]. In well-posed systems, the Laplace transform
of the input-output map gives the transfer function of the system, which plays
an important role in the present paper. For well-posed systems, it becomes more
difficult to determine the generators of the feedback systems, cf. [28]. However, in
our arguments we can avoid to use a precise description of these operators. For
well-posed systems exponential stabilizability and detectability was discussed in
many papers, see, e.g., [9], [12], [13], [20], [21], [29], and the references therein.](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-16-320.jpg)
![Polynomial Internal and External Stability 3
In this paper we will weaken the exponential stability of the feedback sys-
tem in the above concepts to polynomial stability. Here the feedback systems are
described by equations for the resolvents of the generators of given and the feed-
back semigroup which are coupled via a perturbation term involving the feedback,
see Definitions 3.1 and 3.1. In the study of the resulting concepts of polynomial
stabilizability and detectability we pursue two main questions, also treated in the
above papers.
We show that a system possesses these properties if it can be decomposed
into a polynomial stable and an observable part, see Theorem 4.6 and 4.7. In the
exponential case, such results are often called pole-assignment if the stable part
has a finite-dimensional complement. Actually one can derive exponential stabi-
lizability from much weaker concepts (optimizability or the finite cost condition),
see [9] or [29]. So far it is not clear whether such implications hold for the natural
analogues of these concepts to the polynomial setting. Moreover, it is known that
optimizability can be characterized by decompositions as above if the resolvent
set of the generator contains a strip around iR, see [12] or [21]. In the polynomial
setting one here has to fight against the fact that the spectrum may approach the
imaginary axis at infinity. So far we only have partial results in this context, not
treated below.
The main part of our results is devoted to the relationship between poly-
nomial stability of the given semigroup and polynomial estimates on the trans-
fer function of the system. It is known that A generates an exponentially stable
semigroup if (and only if) the system (A, B, C) is exponential stabilizable and
detectable and its transfer function is bounded on the right half-plane, see [20]
and also [29] for an extension to the concepts of optimizability and estimatibil-
ity. (Note that the ‘only if’ implication is easily shown with 0 feedbacks.) The
boundedness of the transfer function is called external stability. In Theorem 4.3
we extend these results to our setting, thus requiring polynomial stabilizability
and detectability and that the transfer function grows at most polynomially as
| Im λ| → ∞. (The latter condition may be called polynomial external stability.) If
the involved semigroups are bounded, we then obtain polynomial stability of the
order one expects, i.e., the sum of the orders in the assumption. The proofs are
based on various estimates and manipulations of formulas connecting resolvents,
the transfer functions and their variants. We further use the results polynomial
stability from [6] and [7] mentioned above.
If the given semigroup is not known to be bounded, then the available the-
ory on polynomial stability does not give the above-indicated convergence order.
However, in applications one can often check the boundedness of a semigroup by
the dissipativity of its generator, possibly for an equivalent norm. Similarly one
can characterize well-posed systems with energy dissipation (so-called scattering
passive systems), see, e.g., [22]. Besides the given semigroup, here also the trans-
fer function is contractive which leads to an improvement of our main result for
scattering passive systems, see Corollary 4.4. In general, not much is known on
the preservation of boundedness under perturbations. In Theorem 5 of the recent](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-17-320.jpg)
![4 E.M. Ait Benhassi, S. Boulite, L. Maniar and R. Schnaubelt
paper [19] one finds a result which requires smallness of the perturbations as maps
into spaces between D(A) and X. In Proposition 4.5 we show the boundedness in
the framework of the present paper. Our approach is based on a characterization
of bounded semigroups in terms of L2
-norms of the resolvents of A and A∗
due to
[10], see Proposition 2.4.
In the next section we discuss the background on polynomial stability and
well-posed systems. In Section 3 we introduce polynomial stabilizability and de-
tectability and establish several basic estimates. The last section contains our main
results on external polynomial stability and on sufficient criteria for polynomial
stabilizability and detectability.
2. Polynomial stability and well-posed systems
We first discuss polynomially stable semigroups. Throughout T (·) denotes a C0-
semigroup on a Banach space X with generator A. There are numbers ∈ R and
M ≥ 1 such that T (t) ≤ Met
for all t ≥ 0. The infimum of these numbers
is denoted by ω0(A). The semigroup is called bounded if T (t) ≤ M for all t ≥ 0.
We fix some ω ω0(A). It is well known that then the fractional powers
(ω − A)β
exist for β ∈ R. They are bounded operators for β ≤ 0 and closed ones
for β 0. The domain Xβ of (ω − A)β
for β 0 is endowed with the norm given
by xβ = (ω − A)β
x. The fractional powers satisfy the power law and coincide
with usual powers for β ∈ Z. In particular, (ω − A)−β
is the inverse of (ω − A)β
for all β ∈ R. We next recall a definition from [5].
Definition 2.1. A C0-semigroup T (·) is called polynomially stable (of order α 0)
if there is a constant α 0 such that
T (t)(ω − A)−α
≤ ct−1
for all t ≥ 1.
(Here and below, we write c 0 for a generic constant.) Note that a larger
order α means a weaker convergence property. Due to Proposition 3.1 of [5], a
bounded C0-semigroup T (·) is polynomially stable of order α 0 if and only if
T (t)(ω − A)−αγ
≤ c(γ) t−γ
, t ≥ 1, (2.1)
for all/some γ 0. (There is also a partial extension to general C0-semigroups.)
Combined with (2.1), Proposition 3 of [6] yields the following necessary con-
dition for polynomial stability of bounded C0-semigroups. Here we set
C± = {λ ∈ C
Re λ ≷ 0} and Cr = r + C+ for r ∈ R.
Proposition 2.2. Let T (·) be a bounded C0-semigroup which is polynomially stable
of order α 0. Then the spectrum σ(A) of A belongs to C− and its resolvent is
bounded by
R(λ, A) ≤ c (1 + |λ|)α
for all λ ∈ C+. (2.2)](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-18-320.jpg)
![Polynomial Internal and External Stability 5
Due to Lemma 3.2 in [14], the estimate (2.2) is true if and only if
R(λ, A)(ω − A)−α
≤ c for all λ ∈ C+. (2.3)
If one drops the boundedness assumption, the above result still holds with an
epsilon loss in the exponent in the right-hand side of (2.2) by Proposition 3.3 of
[5] and (2.3). We further note that condition (2.2) implies the inclusion
{λ ∈ σ(A)
Re λ ≥ −δ} ⊂ {λ ∈ C−
| Im λ| ≥ c(− Re λ)−1/α
}
for some c, δ 0, see Proposition 3.7 of [5].
The next result from [7] provides the important converse of the above propo-
sition for bounded semigroups on a Hilbert space, see Theorem 2.4 of [7].
Theorem 2.3. Let T (·) be a bounded C0-semigroup on a Hilbert space X such that
σ(A) ⊂ C− and (2.2) holds for all λ ∈ iR. Then T (·) is polynomially stable of
order α 0.
For general Banach spaces X, in Theorem 5 in [6] this result was shown up
to a logarithmic factor in the estimate in semigroup, see also [5], [15] and [17].
The paper [7] gives an example where this logarithmic correction actually occurs.
Without assuming its boundedness, the semigroup is still polynomially stable if
a holomorphic extension of R(λ, A)(ω − A)−α
satisfies (2.3), but here one only
obtains the stability order 2α + 1 + for any 0, see Proposition 3.4 of [5].
The proof of Theorem 2.3 is based on the following characterization of the
boundedness of C0-semigroups on Hilbert spaces, see Theorem 2 in [10] and also
Lemma 2.1 in [7].
Proposition 2.4. Let A generate the C0-semigroup T (·) on the Hilbert space X.
The semigroup is bounded if and only if C+ ⊂ ρ(A) and
sup
r0
r
R
R(r + iτ, A)x2
+ R(r + iτ, A∗
)x2
dτ ≤ c x2
for each x ∈ X.
We now turn our attention to the concept of well-posed systems. From now
on, X, U and Y are always Hilbert spaces, A generates the C0-semigroup T (·) on X
and ω ω0(A). Let X−1 be the completion of X with respect to the norm given by
x−1 = R(ω, A)x. We sometimes write XA
−1 instead of X−1 to stress that this
extrapolation space depends on A. The operator A has a unique extension A−1 ∈
B(X, X−1) which generates a C0-semigroup given by the continuous extension
T−1(t) ∈ B(X−1) of T (t), t ≥ 0. We often omit the subscript −1 here. One can
define such a space for each linear operator with non-empty resolvent set. Recall
that we have set X1 = D(A).
A bounded linear (observation) operator B : U −→ X−1 is called admissible
for A (or the system (A, B, −) is called admissible) if the integral
Φtu :=
t
0
T (t − s)Bu(s) ds](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-19-320.jpg)
![6 E.M. Ait Benhassi, S. Boulite, L. Maniar and R. Schnaubelt
belongs to X for all u ∈ L2
(0, t; U) and some t 0. (The integral is initially
defined in X−1.) By Proposition 4.2.2 in [24], this property then holds for all t ≥ 0
and Φt ∈ B(L2
(0, t; U), X). Moreover, these operators are exponentially bounded,
see Proposition 4.4.5 in [24].
A bounded linear (control) operator C : X1 −→ Y is called admissible for A
(or the system (A, −, C) is called admissible) if the map
Ψtx := CT (·)x, x ∈ X1,
has a bounded extension in B(X, L2
(0, t; Y )) for some t 0. Propositions 4.2.3
and 4.3.3 in [24] show that this fact then holds for all t 0 and that the extensions
are exponentially bounded. We still denote the extension by Ψt. One can extend
an admissible observation operator C to the map CΛ given by
CΛx = lim
λ→∞
CλR(λ, A)x
with domain D(CΛ) = {x ∈ X
this limit exists in Y }. For each x ∈ X we have
T (s)x ∈ D(CΛ) for a.e. s ≥ 0 and Ψtx = CΛT (·)x a.e. on [0, t] for all t 0 by,
e.g., (5.6) and Proposition 5.3 in [28].
Theorem 4.4.3 of [24] shows that an operator B ∈ B(U, X−1) is admissible
for A if and only if its adjoint B∗
∈ B(D(A∗
), U) is admissible for A∗
. Here we
recall that X−1 is the dual space of D(A∗
), if considered as a Banach space, see,
e.g., Proposition 2.10.2 in [24].
Let system (A, B, C) be a system with a generator A and admissible control
and observation operators B and C. One says that (A, B, C) is well posed if there
are bounded linear operators Ft : L2
(0, t; U) −→ L2
(0, t; Y ) such that
Fτ+tu =
Fτ u1 on [0, τ],
Ftu2 + ΨtΦτ u1 on [τ, τ + t]
for all t, τ ≥ 0 and u ∈ L2
(0, τ + t; U), where u = u1 on (0, τ) and u = u2 on
(τ, τ + t), see [27]. Also these (input-output) operators are exponentially bounded
by Proposition 4.1 of [27].
One can introduce versions of the maps Ψt and Ft on the time interval R+
using L2
loc spaces. We denote these extensions by Ψ and F respectively. For x0 ∈ X
and u ∈ L2
loc(R+, U) the output of the well-posed system (A, B, C) is then given
by y = Ψx0 + Fu. In [27] it was shown that the Laplace transform ŷ of y satisfies
ŷ(λ) = C(λ − A)−1
x0 + G(λ)û(λ)
for all λ ∈ Cω, where G : Cω → B(U, Y ) is a bounded analytic function. It
satisfies G
(λ) = −CR(λ, A)2
B and it is thus determined by A, B and C up to an
additive constant. (See, e.g., Theorem 2.7 in [22].) We call G the transfer function
of (A, B, C).
Set Z = D(A)+R(ω, A−1)BU and endow it with the norm zZ given by the
infimum of all x1 + R(ω, A−1)Bv with z = x + R(ω, A−1)Bv, x ∈ D(A) and](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-20-320.jpg)
![Polynomial Internal and External Stability 7
v ∈ U. Theorem 3.4 and Corollary 3.5 of [22] then yield an extension C ∈ L(Z, U)
of C such that the transfer function is represented as
G(λ) = CR(λ, A−1)B + D, λ ∈ Cω, (2.4)
for a feedthrough operator D ∈ L(U, Y ). Hence, the operators CR(λ, A−1)B are
uniformly bounded on Cω.
This representation of G is not unique in general since D(A) need not to be
dense in Z. Under the additional assumption of regularity, one can replace here C
by CΛ (possibly for a different D), see Theorem 5.8 in [27] and also Theorem 4.6
in [22] for refinements. We will not use regularity below.
3. Polynomial stabilizability and detectability
In this section we introduce our new concepts and establish their basic properties.
We start with the main definitions.
Definition 3.1. The admissible system (A, B, −) is polynomially stabilizable (of or-
der α 0) if there exists a generator ABF of a polynomially stable C0-semigroup
TBF (·) on X (of order α 0) and an admissible observation operator F ∈
L(D(ABF ), U) of ABF such that
R(λ, ABF ) = R(λ, A) + R(λ, A)BFR(λ, ABF ) (3.1)
for all Re λ max{ω0(A), ω0(ABF )}.
Definition 3.2. The admissible system (A, −, C) is polynomially detectable (of order
α 0) if there exists a generator AHC of a polynomially stable C0-semigroup
THC(·) (of order α 0) and an admissible control operator H ∈ L(Y, XAHC
−1 ) of
AHC such that
R(λ, AHC ) = R(λ, A) + R(λ, (AHC )−1)HCR(λ, A) (3.2)
for all Re λ max{ω0(A), ω0(AHC )}.
Here F, resp. H, plays the role of a feedback. These definitions are inspired
by the Definition 3.2 in [12] for the exponentially stable case. For this case, in,
e.g., [29] concepts of exponential stabilizability or detectability were used which
are (at least formally) a bit stronger than those in [12], cf. Remark 3.3(b). In our
context, one could also include the boundedness of the feedback semigroup TBF (·)
or THC (·) in the above definitions since the theory of polynomial stability works
much better in the bounded case, as seen in the previous section. Instead, we make
additional boundedness assumptions in some of our results. In applications one can
check the boundedness or TBF (·) or THC (·) by showing that the generators ABF
or AHC are dissipative, respectively, where one may use their representation given
in the next remark.](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-21-320.jpg)
![8 E.M. Ait Benhassi, S. Boulite, L. Maniar and R. Schnaubelt
Remark 3.3. (a) Let (A, B, −), (ABF , −, F), (A, −, C) and (AHC , H, −) be admis-
sible. Proposition 4.11 in [13] (with β = γ = 1 and b = c = 0) then shows that the
equations (3.1) and (3.2) are equivalent to
TBF (t)x = T (t)x +
t
0
T (t − s)BFΛTBF (s)x ds = T (t)x + ΦtFΛTBF (·)x, (3.3)
THC(t)x = T (t)x +
t
0
THC(t − s)HCΛT (s)x ds (3.4)
for all t ≥ 0 and x ∈ X, respectively.
(b) Applying λ − A−1 to (3.1), we see that ABF is a restriction of the part
(A−1 +BF)|X of A−1 +BF in X. Similarly, multiplication of (3.2) by λ−AHC,−1
leads to A ⊂ (AHC,−1 − HC)|X. See Proposition 6.6 in [28]. We note that in
[29] exponential stabilizability and detectability was defined in such a way that
ABF = (A−1 + BFΛ)|X and AHC = (A−1 + CHΛ)|X.
(c) The system (A, B, −) is polynomially stabilizable of order α 0 (with
feedback F) if and only if (A∗
, −, B∗
) is polynomially detectable of order α 0
(with feedback H = F∗
). Moreover, the semigroups of the feedback systems are
dual to each other.
(d) Let L be a closed operator with ∅ = Λ ⊂ ρ(L) and Ω ⊃ Λ be connected.
If R(·, L) has a holomorphic extension Rλ to Ω, then Ω ⊂ ρ(L) and Rλ = R(λ, L)
for every λ ∈ Ω. (See Proposition B5 in [3].)
In a sequence of lemmas we relate the growth properties of several operators
arising in (3.1) or (3.2). We use the spectral bound s(L) = sup{Re λ
λ ∈ σ(L)} ∈
[−∞, ∞] for a closed operator L, where sup ∅ = −∞
Lemma 3.4. Let C ∈ B(X1, Y ) and B ∈ B(U, X−1) be admissible observation and
control operators for A, respectively and let
R(r + iτ, A) ≤ c |τ|α
(3.5)
for some r s(A) and α 0 and all |τ| ≥ 1. We then obtain the estimates
CR(r + iτ, A) ≤ c |τ|α
and R(r + iτ, A)B ≤ c |τ|α
for all |τ| ≥ 1. Moreover, if (A, B, C) is also well posed, we have
CR(r + iτ, A)B ≤ c |τ|α
for all |τ| ≥ 1. Here the constants are uniform for r in bounded intervals.
Proof. Let λ = r + iτ and μ = ω + iτ for τ ∈ R and some ω max{0, ω0(A)}.
The resolvent equation yields
CR(λ, A) = CR(μ, A) + (ω − r)CR(μ, A)R(λ, A). (3.6)
Let x ∈ D(A). Since the resolvent is the Laplace transform of T (·), from the
admissibility of C and exponential bound of T (·) we deduce
CR(μ, A)x2
≤
∞
0
e− ω
2 t
e− ω
2 t
CT (t)x dt
2
≤ c
∞
0
e−ωt
CT (t)x2
dt (3.7)](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-22-320.jpg)
![Polynomial Internal and External Stability 9
≤ c
∞
n=0
e−ωn
CT (·)T (n)x2
L2(0,1;Y ) ≤ c
∞
n=0
e−ωn
T (n)x2
≤ c x2
.
By density, the formulas (3.5), (3.6) and (3.7) imply
CR(λ, A) ≤ c + c |τ|α
≤ c |τ|α
for |τ| ≥ 1. The second asserted inequality then follows by duality because B∗
is
an admissible observation operator for A∗
and R(λ, A)B = B∗
R(λ, A∗
). For
the final claim, we start from the equation
CR(λ, A)B = CR(μ, A)B + (ω − r)CR(μ, A)R(λ, A)B
for λ = r + iτ, μ = ω + iτ, τ ∈ R and some ω max{0, ω0(A)}. As noted in the
previous section, CR(μ, A)B : U → Y is uniformly bounded. The third assertion
now is a consequence of the two previous ones.
In the next lemma we deduce resolvent estimates for A from those for ABF .
Lemma 3.5. Let B ∈ L(U, X−1) be an admissible control operator for A. Assume
that there exist a generator ABF of a C0-semigroup TBF (·) on X and an admissible
observation operator F ∈ L(D(ABF ), U) of ABF such that (3.1) holds. Assume
that
R(λ, ABF ) ≤ c (1 + |λ|α
)
for r Re λ ≤ r + δ and some r ≥ s(ABF ), δ 0, α ≥ 0. Suppose that R(λ, A)B
has a holomorphic extension RB
λ to Cr satisfying
RB
λ ≤ c (1 + |λ|β
)
for r Re λ ≤ r + δ and some β ≥ 0. Then R(·, A) can be extended to a neigh-
borhood of Cr, and we obtain
R(λ, A) ≤ c (1 + |λ|α+β
) (3.8)
for r ≤ Re λ ≤ r + δ. Moreover, (3.1) holds on Cr. If r = 0, then T (·) is polyno-
mially stable with order 2(α + β) + 1 + η for any η 0.
Proof. By the assumption, (3.1) and Remark 3.3, the resolvent R(·, A) has the
extension
R(λ, A) = R(λ, ABF ) − RB
λ FR(λ, ABF )
to λ ∈ Cr. Lemma 3.4 and the assumption then imply that
R(λ, A) ≤ c (1 + |λ|α+β
)
for r Re λ ≤ r + δ. A standard power series argument allows us to extend this
inequality to λ ∈ Cr and to deduce that a neighborhood of Cr belongs to ρ(A).
The uniqueness of the holomorphic extension now yields that RB
λ = R(λ, A)B on
Cr and that (3.1) holds on Cr. The last assertion then follows from estimate (3.8)
and Propositions 3.4 and 3.6 in [5].
The next result is proved in the same manner as the above lemma.](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-23-320.jpg)
![Polynomial Internal and External Stability 11
a) The resolvent R(·, A) can be extended to a neighborhood of C+ and
R(λ, A) ≤ cε (1 + |λ|α+β+ε
) (4.1)
for 0 ≤ Re λ ≤ δ and every ε 0. If TBF (·) is bounded, we can choose ε = 0.
b) The semigroup T (·) is polynomially stable. If T (·) is also bounded, then it is
polynomially stable of order α + β + ε. If in addition TBF (·) is bounded, we
can take ε = 0.
Proof. a) Propositions 3.3 and 3.6 in [5] imply that σ(ABF ) ⊂ C− and
R(λ, ABF ) ≤ cε(1 + |λ|α+ε
)
holds for Re λ ≥ 0 and every ε 0. Using Lemma 3.5, we infer σ(A) ⊂ C− and
(4.1). If TBF (·) is bounded, we can use Proposition 2.2 instead of the results from
[5] and obtain the above estimates with ε = 0.
b) Proposition 3.4 of [5] and (4.1) imply the polynomial stability of T (·). If
also T (·) is bounded, it is polynomially stable of order α+β+ε due to Theorem 2.3
and (4.1).
By duality, the above proposition implies the next one for the observation
system (A, −, C).
Proposition 4.2. Let (A, −, C) be admissible and polynomially detectable of order
α 0. Assume that CR(·, A) has a holomorphic extension to C+ which is bounded
by c (1 + |λ|β
) for 0 Re λ ≤ δ and some β ≥ 0. The following assertions hold.
a) The resolvent R(·, A) can be extended to a neighborhood of C+ and estimate
(4.1) holds for every ε 0. If THC(·) is bounded, we can take ε = 0.
b) The semigroup T (·) is polynomially stable. If T (·) is also bounded, then it is
polynomially stable of order α + β + ε. If in addition THC (·) is bounded, we
can take ε = 0.
We now can state our main result which uses the full system (A, B, C) and
the transfer function G.
Theorem 4.3. Let (A, B, C) be a well-posed system which is polynomially stabiliz-
able of order α 0 and polynomially detectable of order β 0. Assume that G has
a holomorphic extension to C+ which is bounded by c (1 + |λ|γ
) for 0 Re λ ≤ δ
and some γ ≥ 0 and δ 0. The following assertions hold.
a) The extension C of C is an admissible observation operator for ABF , σ(A) ⊂
C−, and
R(λ, A) ≤ cε(1 + |λ|α+β+γ+ε
)
for 0 Re λ ≤ δ and all ε 0. If TBF (·) is bounded, we can take ε = 0.
b) The semigroup T (·) is polynomially stable. If T (·) is bounded, then it is poly-
nomially stable of order α + β + γ + ε. If in addition TBF (·) is bounded, we
can take ε = 0.](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-25-320.jpg)
![12 E.M. Ait Benhassi, S. Boulite, L. Maniar and R. Schnaubelt
Proof. a) Due to (3.1) and (2.4), we have D(ABF ) ⊂ Z and
CR(λ, ABF ) = CR(λ, A) + CR(λ, A)BFR(λ, ABF ),
CR(λ, ABF ) = CR(λ, A) + G(λ)FR(λ, ABF ) − DFR(λ, ABF ) (4.2)
for Re λ max{ω0(A), ω0(ABF )}. Taking the inverse Laplace transform of this
equation, we define
ΨBF x := L−1
(CR(·, ABF )x) = Ψx + FFTBF (·)x − DFTBF (·)x (4.3)
for x ∈ D(ABF ). By assumption, ΨBF : X −→ L2
loc(R+, Y ) is continuous. For
τ ≥ 0 and x ∈ D(ABF ), the properties of a well-posed system and (3.3) yield
ΨBF x(· + τ) = ΨT (τ)x + FFTBF (·)TBF (τ)x + ΨΦτ FTBF (·)x
− DFTBF (·)TBF (τ)x
= ΨTBF (τ)x + FFTBF (·)TBF (τ)x − DFTBF (·)TBF (τ)x
= ΨBF TBF (τ)x.
As a result, (ΨBF , TBF ) is an observation system in the sense of [26] or Sec-
tion 4.3 in [24]. The proof of Theorem 3.3 of [26] and (4.3) thus show that ΨBF x =
CTBF (·)x for x ∈ D(ABF ) and the admissible control operator C ∈ L(D(ABF ), Y )
for ABF given by
Cx =
ΨBF (λ)(λ − ABF )x = CR(λ, ABF )(λ − ABF )x = Cx for x ∈ D(ABF );
i.e., ΨBF x = CTBF (·)x for x ∈ D(ABF ). Proposition 3.4 of [5] and Lemma 3.4
then yield
CR(λ, ABF ) ≤ c (1 + |λ|α+ε
) and FR(λ, ABF ) ≤ c (1 + |λ|α+ε
)
for Re λ ≥ 0 and any ε 0. If TBF (·) is bounded, we can use Proposition 2.2
instead of the results in [5] and derive these estimates with ε = 0. By means of
(4.2) and the bound on G, we now extend CR(·, A) (using the same symbol) to
C+ and obtain
CR(λ, A) ≤ c (1 + |λ|α+γ+ε
)
for 0 Re λ ≤ δ. Proposition 4.2 then gives
R(λ, A) ≤ cε(1 + |λ|α+β+γ+ε
)
for 0 Re λ ≤ δ and all ε 0, where we can take ε = 0 if TBF (·) is bounded.
b) Proposition 3.4 of [5] and part a) imply the polynomial stability of T (·). If
T (·) is bounded, it is polynomially stable of order α+β +γ +ε due to Theorem 2.3
and part a), where we can take ε = 0 if TBF (·) is bounded.
In the above results one obtains the expected stability order of T (·) only if
this semigroup is bounded. This property automatically holds in the important
case of a scattering passive system (A, B, C); i.e., if we have
y2
L2(0,t;Y ) + x(t)2
≤ u2
L2(0,t;U) + x02](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-26-320.jpg)
![Polynomial Internal and External Stability 13
for all u ∈ L2
(0, t; U), x0 ∈ X and t ≥ 0, where x(t) = T (t)x0 + Φtu is the state
and y = Ψx0 + Fu is the output of (A, B, C). This class of systems has been
characterized and studied in, e.g., [22]. In this case T (t) and G(λ) are contractions
for t ≥ 0 and λ ∈ C+ by Proposition 7.2 and Theorem 7.4 of [22].
Corollary 4.4. Let (A, B, C) be a scattering passive system which is polynomially
stabilizable of order α 0 and polynomially detectable of order β 0. Then T (·)
is polynomially stable of order α + β + ε for each ε 0. We can take ε = 0 if
TBF (·) is bounded.
Proposition 2.4 yields another sufficient condition for the boundedness of T (·)
in the framework of the first two propositions of this section.
Proposition 4.5. Assume that the assumptions of both Propositions 4.1 and 4.2
hold for some α 0 and for β = 0. Let TBF (·) and THC(·) be bounded. Then T (·)
is bounded, and hence polynomially stable of order α 0.
Proof. Definitions 3.1 and 3.2 yield
R(r + iτ, A)x = R(r + iτ, ABF )x − R(r + iτ, A)BFR(r + iτ, ABF )x, (4.4)
R(r+iτ, A∗
)x = R(r + iτ, A∗
HC )x − R(r + iτ, A∗
)C∗
H∗
R(r+iτ, A∗
HC)x (4.5)
for all r max{ω0(A), 0}, τ ∈ R and x ∈ X. We can extend these equations to
r 0 using the bounded extensions of R(λ, A)B and R(λ, A∗
)C∗
= (CR(λ, A))∗
which are provided by our assumption. Since TBF (·) and THC(·) are bounded,
Lemma 3.7 implies that the terms on the right-hand sides belong to L2
(R, X) as
functions in τ, with norms bounded by cr−1/2
x. Employing Proposition 2.4, we
then deduce the boundedness of T (·) from (4.4) and (4.5). The final assertion now
follows from Proposition 4.1.
We finally present sufficient conditions for polynomial stabilizability and for
polynomial detectability by means of a decomposition into a polynomial stable
and an observable part. An admissible system (A, B, −) is called null controllable
in finite time if for each initial value x0 ∈ X there is a time τ 0 and a control
u ∈ L2
(0, τ; U) such that x(τ) = T (τ)x0 + Φτ u = 0. We further note that one can
extend an operator S to X−1 if it commutes with T (t) for all t ≥ 0 since then
SR(ω, A) = R(ω, A)S.
Theorem 4.6. Let (A, B, −) be admissible and let P2
= P ∈ B(X) satisfy T (t)P =
PT (t) for all t ≥ 0. Set Xs = PX, Xu = (I −P)X, Ts(t) = T (t)P, Au = (I −P)A
and Bu = (I − P)B. Assume that
(i) the C0-semigroup Ts(·) is polynomially stable of order α 0 on Xs and
(ii) the system (Au, Bu, −) is null controllable in finite time on Xu.
Then the system (A, B, −) is polynomially stabililizable of order α 0.
Proof. First observe that Tu(·) is the C0-semigroup on Xu generated by Au and
that Bu is admissible for Au. Due to (ii), for each x0 ∈ Xu there is a time τ 0 and
a control u ∈ L2
(0, τ; U) such that xu(τ) = Tu(τ)x0 + (I − P)Φτ u = 0. Extending](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-27-320.jpg)
![14 E.M. Ait Benhassi, S. Boulite, L. Maniar and R. Schnaubelt
xu and u by 0 to (τ, ∞), we see that the system (Au, Bu, −) is optimizable in the
sense of Definition 3.1 in [29]. Propositions 3.3 and 3.4 of [29] (or Theorem 2.2 of
[9]) then give an operator Fu which satisfies the conditions of Definition 3.1 where
TBuFu (·) is even exponentially stable, i.e., ω0(ABuFu ) 0. We thus have
R(λ, ABuFu ) = R(λ, Au) + R(λ, Au)BuFuR(λ, ABuFu ) (4.6)
for all Re λ max(ω0(A), ω0(ABuFu )). We now set
F =
0
Fu
and ABF :=
As 0
0 ABuFu
.
It is then straightforward to check that these operators fulfill the conditions of
Definition 3.1.
The next result follows by duality from Theorem 4.6.
Theorem 4.7. Let (A, −, C) be admissible and let P2
= P ∈ B(X) satisfy T (t)P =
PT (t) for all t ≥ 0. Set Xs = PX, Xu = (I −P)X, Ts(t) = T (t)P, Au = (I −P)A
and Cu = C(I − P). Assume that
(i) the C0-semigroup Ts(·) is polynomially stable of order α 0 on Xs and
(ii) the system (A∗
u, C∗
u, −) is null controllable in finite time on Xu.
Then the system (A, −, C) is polynomially detectable of order α 0.
Remark 4.8. The results of Theorem 4.6 and 4.7 also hold if we replace the con-
dition (ii) by (ii)
: The system (Au, Bu, −) (resp., (A∗
u, C∗
u, −)) is polynomially
stabilizable of order α.
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[1] F. Alabau, P. Cannarsa and V. Komornik, Indirect internal stabilization of weakly
coupled evolution equations. J. Evol. Equ. 2 (2002), 127–150.
[2] K. Ammari and M. Tucsnak, Stabilization of second-order evolution equations by a
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angular domains. Math. Res. Lett. 14 (2007), 35–47.](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-28-320.jpg)
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(1989), 17–43.](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-29-320.jpg)
![16 E.M. Ait Benhassi, S. Boulite, L. Maniar and R. Schnaubelt
[27] G. Weiss, Transfer functions of regular linear systems. Part I: Characterization of
regularity, Trans. Amer. Math. Soc. 342 (1994), 827–854.
[28] G. Weiss, Regular linear systems with feedback. Math. Control Signals Systems 7
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systems. SIAM J. Control Optim. 39 (2000), 1204–1232.
El Mustapha Ait Benhassi and Lahcen Maniar
Cadi Ayyad University
Faculty of Sciences Semlalia
B.P. 2390
Marrakesh, Morocco
e-mail: m.benhassi@ucam.ac.ma
maniar@ucam.ac.ma
Said Boulite
Hassan II University
Faculty of Sciences Ain Chock
B.P. 5366 Maarif
20100 Casablanca, Morocco
e-mail: s.boulite@fsac.ac.ma
Roland Schnaubelt
Department of Mathematics
Karlsruhe Institute of Technology
D-76128 Karlsruhe, Germany
e-mail: schnaubelt@kit.edu](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-30-320.jpg)
![Operator Theory:
Advances and Applications, Vol. 250, 17–29
Minimal Primal Ideals in the Multiplier
Algebra of a C0(X)-algebra
R.J. Archbold and D.W.B. Somerset
Abstract. Let A be a stable, σ-unital, continuous C0(X)-algebra with sur-
jective base map φ : Prim(A) → X, where Prim(A) is the primitive ideal
space of the C∗
-algebra A. Suppose that φ−1
(x) is contained in a limit set in
Prim(A) for each x ∈ X (so that A is quasi-standard). Let CR(X) be the ring
of continuous real-valued functions on X. It is shown that there is a homeo-
morphism between the space of minimal prime ideals of CR(X) and the space
MinPrimal(M(A)) of minimal closed primal ideals of the multiplier algebra
M(A). If A is separable then MinPrimal(M(A)) is compact and extremally
disconnected but if X = βN N then MinPrimal(M(A)) is nowhere locally
compact.
Mathematics Subject Classification (2010). Primary 46L05, 46L08, 46L45; Sec-
ondary 46E25, 46J10, 54C35.
Keywords. C∗
-algebra, C0(X)-algebra, multiplier algebra, minimal prime
ideal, minimal primal ideal, primitive ideal space, quasi-standard.
1. Introduction
Let A be a C∗
-algebra with multiplier algebra M(A) [10] and with primitive ideal
space Prim(A). The ideal structure of M(A) has been widely studied, and is typi-
cally much more complicated than that of A, see for example [1], [13], [21], [25],[27].
One approach, which the authors used in an earlier paper [7], is to endow A with a
C0(X)-structure (this can always be done, sometimes in many different ways). Let
A be a σ-unital C0(X)-algebra (defined below) with base map φ : Prim(A) → X,
and let Xφ denote the image of Prim(A) under φ. The authors showed that there
is a map from the lattice of z-ideals of CR(Xφ) into the lattice of closed ideals
of M(A), and that this map is injective if A is stable [7, Theorem 3.2]. If Xφ is
infinite then z-ideals generally exist in great profusion – for example, CR(R) has
uncountable chains of prime z-ideals associated with each point of R [22], [26] –
c
Springer International Publishing Switzerland 2015](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-31-320.jpg)
![18 R.J. Archbold and D.W.B. Somerset
so this yields a vast multiplicity of closed ideals in M(A) and indicates something
of the complexity of Prim(M(A)) [7, Theorem 5.3].
The most studied z-ideals are the minimal prime ideals and in this note
we consider the image of the space of minimal prime ideals of CR(Xφ) under
the injective map. We show in Theorem 3.4 that if A is stable, σ-unital, and
quasi-standard (defined below) then the image of the space of minimal prime
ideals is precisely MinPrimal(M(A)), the space of minimal closed primal ideals
of M(A) (see below). It follows that MinPrimal(M(A)) is totally disconnected
and countably compact (Corollary 4.1). If A is also separable – for example if
A equals C[0, 1] ⊗ K(H) (where K(H) is the algebra of compact operators on a
separable infinite-dimensional Hilbert space) – then MinPrimal(M(A)) is compact
and extremally disconnected (Corollary 4.3).
All ideals in this paper will be two-sided, but not necessarily closed unless
stated to be so. An ideal J in a C∗
-algebra A is primal if whenever I1, . . . , In
is a finite collection of ideals of A with the product I1 . . . In = {0} then Ii ⊆ J
for at least one i ∈ {1, . . . , n}. An equivalent definition, when J is closed, is that
the hull of J should be contained in a limit set in Prim(A) [3, Proposition 3.2].
Every primitive ideal is prime and hence primal. Each closed primal ideal of a
C∗
-algebra A contains one or more minimal closed primal ideals [2, p. 525]. The
space of minimal closed primal ideals with the τw-topology (defined in Section 3)
is denoted MinPrimal(A). This Hausdorff space is often identifiable in situations
where the primitive ideal space is non-Hausdorff and highly complicated. Indeed,
the multiplier algebras considered in this paper are a case in point.
A C∗
-algebra A is said to be quasi-standard if the relation ∼ of inseparability
by disjoint open sets is an open equivalence relation on Prim(A) [5]. This condition
is a wide generalisation of the special case where Prim(A) is Hausdorff. Examples
include, in the unital case, von Neumann and AW∗
-algebras, local multiplier alge-
bras of C∗
-algebras [29], and the group C∗
-algebras of amenable discrete groups
[17]; and in the non-unital case, many other group C∗
-algebras, see [4]. A basic
non-unital example, however, is simply A = C0(X) ⊗ K(H), where X is a locally
compact Hausdorff space, and even in this case the ideal structure of M(A) is not
well understood, see [20], [7]. The connection between quasi-standard C∗
-algebras
and C0(X)-algebras is explained in Lemma 2.1 and the remarks preceding it.
The structure of the paper is that in Section 2, we set up some machinery;
in Section 3, we prove the main homeomorphism result; and in Section 4, we give
some applications.
2. Preliminaries
First we collect the information that we need on C0(X)-algebras. Recall that a
C∗
-algebra A is a C0(X)-algebra if there is a continuous map φ, called the base
map, from Prim(A), the primitive ideal space of A with the hull-kernel topology,
to the locally compact Hausdorff space X [31, Proposition C.5]. Then Xφ, the](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-32-320.jpg)
![Minimal Primal Ideals in the Multiplier Algebra 19
image of φ in X, is completely regular; and if A is σ-unital, Xφ is σ-compact and
hence normal [7, Section 1]. If φ is an open map then Xφ is locally compact.
For x ∈ Xφ, set Jx = {P ∈ Prim(A) : φ(P) = x}, and for x ∈ X Xφ, set
Jx = A. For a ∈ A, the function x → a + Jx (x ∈ X) is upper semi-continuous
[31, Proposition C.10]. The C0(X)-algebra A is said to be continuous if, for all
a ∈ A, the norm function x → a + Jx (x ∈ X) is continuous. By Lee’s theorem
[31, Proposition C.10 and Theorem C.26], this happens if and only if the base map
φ is open.
An important special case (through which all other cases factor) is when
φ is the complete regularization map φA for Prim(A) [14, Theorem 3.9]. In this
case, the ideals Jx (x ∈ Xφ) are called the Glimm ideals of A, and the set of
Glimm ideals with the complete regularization topology is called Glimm(A). Each
minimal closed primal ideal of A contains a unique Glimm ideal [5, Lemma 2.2]. If
A is quasi-standard then the complete regularization map φA is open [5, Theorem
3.3], so Glimm(A) is locally compact and A is a continuous C0(X)-algebra with
X = XφA = Glimm(A). Furthermore, if A is quasi-standard then each Glimm
ideal of A is actually primal and indeed the topological spaces Glimm(A) and
MinPrimal(A) coincide [5, Theorem 3.3]. It then follows from [3, Proposition 3.2]
that φ−1
A (x) is a maximal limit set in Prim(A) for all x ∈ X. The following result
is closely related to [5, Theorem 3.4].
Lemma 2.1. For a C∗
-algebra A, the following are equivalent:
(i) A is quasi-standard;
(ii) A is a continuous C0(X)-algebra over a locally compact Hausdorff space X
with base map φ such that φ−1
(x) is contained in a limit set in Prim(A) for
all x ∈ Xφ.
When these equivalent conditions hold, there is a homeomorphism ψ : Glimm(A) →
Xφ such that φ = ψ◦φA, where φA is the complete regularization map for A. More-
over, for all x ∈ Xφ, φ−1
(x) is a maximal limit set in Prim(A) and Jx is a minimal
closed primal ideal of A.
Proof. We have seen that (i) implies (ii). Conversely, suppose that (ii) holds. Since
X is a locally compact Hausdorff space, for P, Q ∈ Prim(A), P ∼ Q if and only if
φ(P) = φ(Q). It follows that ∼ is an equivalence relation. Let Y be a non-empty
open subset of Prim(A). Then Y
:= φ−1
(φ(Y )) is the ∼-saturation of Y , and Y
is open since φ is open. Hence ∼ is an open equivalence relation so (i) holds.
When (ii) holds, we have that φ is continuous and open with image Xφ,
and that it factors as φ = ψ ◦ φA, where ψ : Glimm(A) → Xφ is continuous
[14, Theorem 3.9]. Then ψ is surjective, and the limit set hypothesis easily shows
that ψ is injective. Since φ is open and φA is continuous, ψ is open. Thus ψ is a
homeomorphism.
Finally, let x ∈ Xφ and let Ω be a net in Prim(A) whose limit set L contains
φ−1
(x). Since φA is constant on L, φ(L) = {x}. Thus L = φ−1
(x) and φ−1
(x) is a
maximal limit set. It follows from [3, Proposition 3.2] that Jx is a minimal closed
primal ideal of A.](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-33-320.jpg)
![20 R.J. Archbold and D.W.B. Somerset
Now let J be a proper, closed ideal of a C∗
-algebra A. The quotient map qJ : A →
A/J has a canonical extension ˜
qJ : M(A) → M(A/J) such that ˜
qJ (b)qJ (a) =
qJ (ba) and qJ (a) ˜
qJ (b) = qJ (ab) (a ∈ A, b ∈ M(A)). We define a proper, closed
ideal ˜
J of M(A) by
˜
J = ker ˜
qJ = {b ∈ M(A) : ba, ab ∈ J for all a ∈ A}.
Various properties of ˜
J were established in [6, Proposition 1.1]. For example, ˜
J is
the strict closure of J in M(A) and ˜
J ∩ A = J.
The following proposition was proved in [6, Proposition 1.2].
Proposition 2.2. Let A be a C0(X)-algebra with base map φ. Then φ has a unique
extension to a continuous map φ : Prim(M(A)) → βX such that φ(P̃) = φ(P)
for all P ∈ Prim(A). Hence M(A) is a C(βX)-algebra with base map φ and
Im(φ) = clβX(Xφ).
Now let A be a C0(X)-algebra with base map φ and let φ : Prim(M(A)) → βX
be as in Proposition 2.2. For x ∈ βX, we define
Hx =
{Q ∈ Prim(M(A)) : φ(Q) = x},
a closed two-sided ideal of M(A). Thus Hx is defined in relation to (M(A), βX, φ)
in the same way that Jx (for x ∈ X) is defined in relation to (A, X, φ). It fol-
lows that for each b ∈ M(A), the function x → b + Hx (x ∈ βX) is up-
per semi-continuous. If φ is the complete regularization map for Prim(A) and
X = βGlimm(A) then Glimm(M(A)) = {Hx : x ∈ X}; see the comment after [9,
Proposition 4.4].
The next proposition is contained in [7, Proposition 2.3].
Proposition 2.3. Let A be a C0(X)-algebra with base map φ, and set Xφ = Im(φ).
(i) For all x ∈ X, Jx ⊆ Hx ⊆ ˜
Jx and Jx = Hx ∩ A.
(ii) For all b ∈ M(A),
b = sup{b + ˜
Jx : x ∈ Xφ} = sup{b + Hx : x ∈ Xφ}.
In the case when A = C0(X)⊗K(H) ∼
= C0(X, K(H)) and φ : Prim(A) → X is the
homeomorphism such that φ−1
(x) = {f ∈ C0(X) : f(x) = 0} ⊗ K(H) (x ∈ X),
the multiplier algebra M(A) is isomorphic to the C∗
-algebra of bounded, strong∗
-
continuous functions from X to B(H) (the algebra of bounded linear operators on
the Hilbert space H) [1, Corollary 3.5]. Then for x ∈ X,
˜
Jx = {f ∈ M(A) : f(x) = 0}.
On the other hand, by Proposition 2.2 M(A) is a C(βX)-algebra, and for
x ∈ βX,
Hx = {f ∈ M(A) : lim
y→x
f(y) = 0}.](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-34-320.jpg)
![Minimal Primal Ideals in the Multiplier Algebra 21
We shall recall in Theorem 2.4 below that when A is a σ-unital C0(X)-algebra
with base map φ there is an order-preserving map from the lattice of z-ideals of
CR(Xφ) into the lattice of closed ideals of M(A). To describe this map, we give a
brief account of the theory of z-ideals.
Let X be a completely regular topological space and let CR(X) denote the
ring of continuous real-valued functions on X. For f ∈ CR(X), let
Z(f) = {x ∈ X : f(x) = 0},
the zero set of f. We note for later that every zero set clearly arises as the zero
set of a bounded continuous function. A non-empty family F of zero sets of X is
called a z-filter if: (i) F is closed under finite intersections; (ii) ∅ /
∈ F; (iii) each
zero set which contains a member of F belongs to F. Each ideal I ⊆ CR(X) yields
a z-filter Z(I) = {Z(f) : f ∈ I}. An ideal I is called a z-ideal if Z(f) ∈ Z(I)
implies f ∈ I; and if F is a z-filter on X then the ideal I(F) defined by
I(F) = {f ∈ CR(X) : Z(f) ∈ F}
is a z-ideal. There is a bijective correspondence between the set of z-ideals of
CR(X) and the set of z-filters on X, given by I = I(Z(I)) ↔ Z(I).
Now let A be a σ-unital C0(X)-algebra with base map φ, and let u ∈ A be
a strictly positive element. For a ∈ A, set Z(a) = {x ∈ Xφ : a ∈ Jx}. Unless norm
functions of elements of A are continuous on Xφ, Z(a) will not necessarily be a
zero set of Xφ. However, since Z(u) = ∅ and A is closed under multiplication by
Cb
(Xφ), every zero set Z(f) of Xφ arises as Z(a) for the element a = f · u ∈ A
(f ∈ Cb
R(Xφ)). For b ∈ M(A), set Z(b) = {x ∈ Xφ : b ∈ ˜
Jx}. Note that if
b ∈ A then this definition is consistent with the previous one because ˜
Jx ∩ A = Jx
(x ∈ Xφ). It is also useful to note that for b ∈ M(A) and x ∈ Xφ, b ∈ ˜
Jx if and
only if bu ∈ ˜
Jx if and only if bu ∈ Jx. Hence Z(b) = Z(bu).
For a z-filter F on Xφ define Lalg
F = {b ∈ M(A) : ∃Z ∈ F, Z(b) ⊇ Z}, and
let LF be the norm-closure of Lalg
F in M(A). Let b ∈ Lalg
F . Then for a ∈ M(A),
Z(ab) ⊇ Z(b) and Z(ba) ⊇ Z(b), while for a ∈ Lalg
F , Z(a+ b) ⊇ Z(a)∩Z(b). Hence
Lalg
F is an ideal of M(A), so LF is a closed ideal of M(A).
Theorem 2.4. ([7, Theorem 3.2]) Let A be a σ-unital C0(X)-algebra with base map
φ. Suppose that A/Jx is non-unital for all x ∈ Xφ. Let I and J be z-ideals of
CR(Xφ) and suppose that there exists a zero set Z of Xφ such that Z ∈ Z[I] but
Z /
∈ Z[J]. Then LZ[I] ⊆ LZ[J]. Hence the assignment I → LZ[I] defines an order-
preserving injective map L from the lattice of z-ideals of CR(Xφ) into the lattice
of closed ideals of M(A).
To identify what happens to some of the most important z-ideals of CR(Xφ)
under this map, we use the following notation. For x ∈ X, let Mx be the maximal
ideal given by Mx = {f ∈ CR(X) : f(x) = 0}, and let
Ox = {f ∈ CR(X) : x ∈ int(Z(f))}](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-35-320.jpg)
![22 R.J. Archbold and D.W.B. Somerset
where int(Z(f)) denotes the interior of Z(f). Then Mx and Ox are z-ideals, and Ox
is the smallest ideal of CR(X) which is not contained in any maximal ideal other
than Mx. The definitions just given can be extended as follows. For p ∈ βX, let
Mp
= {f ∈ CR(X) : p ∈ clβXZ(f)} and define Op
to be the set of all f ∈ CR(X)
for which clβXZ(f) is a neighbourhood of p in βX. Then for x ∈ X, Mx
= Mx and
Ox
= Ox. The embedding map takes Mx to ˜
Jx and Op
to Hp (and hence Ox to Hx).
Proposition 2.5 ([7, Theorem 4.3]). Let A be a σ-unital C0(X)-algebra with base
map φ.
(i) For x ∈ Xφ, LZ[Mx] = ˜
Jx.
(ii) For p ∈ clβXXφ, LZ[Op] = Hp.
Proposition 2.5 shows that the embedding map of Theorem 2.4 is mainly shedding
light on the lattice of closed ideals of M(A) between ˜
Jx and Hx; see [7, Section 4]
for further discussion. Before presenting a simple example to illustrate Theorem 2.4
and Proposition 2.5, we need further terminology.
A z-filter F on a completely regular space X is said to be prime if Z1 ∪Z2 ∈ F
implies that either Z1 ∈ F or Z2 ∈ F, for zero sets Z1 and Z2. Let PF(X)
denote the set of prime z-filters, and let PZ(X) be the set of prime z-ideals
(recall that an ideal P ⊆ CR(X) is prime if fg ∈ P implies f ∈ P or g ∈ P).
The bijective correspondence between z-ideals and z-filters restricts to a bijective
correspondence j : PZ(X) → PF(X) given by j(P) = {Z(f) : f ∈ P} (see [14,
Chapter 2]). If P ∈ PZ(X) and P ⊆ Mx for some x ∈ X then Ox ⊆ P [14, 4I],
and hence Hx ⊆ LZ[P ] ⊆ ˜
Jx by Proposition 2.5. Every z-ideal of CR(X) is an
intersection of prime z-ideals and the minimal prime ideals of CR(X) are z-ideals
[14, 2.8, 14.7]. The prime ideals containing a given prime ideal form a chain [14,
14.8].
Example. Let X = N ∪ {ω} be the one-point compactification of N and set
A = C(X) ⊗ K(H). Then Mx = Ox for x ∈ N, but Mω = Oω. The assignment
F → PF = {f ∈ CR(X) : Z(f) {ω} ∈ F}
gives a bijection between the family of free ultrafilters on N (every ultrafilter on N
is trivially a z-ultrafilter) and the family of non-maximal prime z-ideals contained
in Mω. Each PF is a minimal prime z-ideal [14, 14G] and we shall see in Section 4
that its image LF under the mapping of Theorem 2.4 is a minimal closed primal
ideal of M(A). The ideal Hω = LZ(Oω) is a Glimm ideal but is not primal.
3. The homeomorphism onto MinPrimal(M(A))
In this section we specialize to the case when A is a σ-unital quasi-standard C∗
-
algebra. We will be assuming that A is canonically represented as a C0(X)-algebra
with the base map φ as the complete regularization map for Prim(A) and with
X = Xφ = Glimm(A). For the main result we will also need to assume that A/Jx
is non-unital for x ∈ X (note that this is automatically satisfied if A is stable).](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-36-320.jpg)
![Minimal Primal Ideals in the Multiplier Algebra 23
The reasons for restricting to quasi-standard C∗
-algebras are twofold. The
first is the fact, already mentioned, that when A is quasi-standard, MinPrimal(A)
and Glimm(A) coincide as sets (and indeed as topological spaces). This has the
implication that, for x ∈ X = Glimm(A), the ideal ˜
Jx is primal in M(A) [6, Lemma
4.5]; and hence there must be minimal closed primal ideals of M(A) lying between
˜
Jx and the Glimm ideal Hx of M(A). But secondly, if A is quasi-standard then
norm functions of elements of A are continuous on Glimm(A), so for a ∈ A, Z(a)
is a zero set of Glimm(A). Furthermore if A is also σ-unital and u ∈ A is a strictly
positive element then, as we have already mentioned, for b ∈ M(A) Z(b) = Z(bu),
so Z(b) is also a zero set of Glimm(A). Thus the elaborate machinery of zero sets
works smoothly for this class of algebras.
For a ring R let Min(R) be the space of minimal (algebraic) primal ideals of
R with the lower topology generated by sub-basic sets of the form
{P ∈ Min(R) : a /
∈ P}
as a varies through elements of R. If R is a commutative ring then an argument of
Krull shows that every minimal primal ideal of R is prime, and Min(R) is the usual
space of minimal prime ideals of R with the hull-kernel topology, see [28] and the
references given there. If P is a minimal prime ideal of CR(X) then P is a z-ideal,
as we have mentioned, so an obvious step is to identify the image of Min(CR(X))
under the embedding map L of Theorem 2.4. We shall show that the embedding
map L carries Min(CR(X)) homeomorphically onto MinPrimal(M(A)) with the
τw-topology (where the τw-topology is defined on MinPrimal(A) by taking sets of
the form {P ∈ MinPrimal(A) : a /
∈ P} (a ∈ A) as sub-basic; see [2, p. 525] where
an equivalent definition is given).
It is convenient to proceed in two stages. In Theorem 3.2 we show that
the assignment P → Lalg
Z[P ] defines a homeomorphism Θ from Min(CR(X)) onto
Min(M(A)). For this theorem we do not require the quotients A/Jx (x ∈ X) to be
non-unital. Then in Theorem 3.4 we show that, if these quotients are non-unital,
the assignment Lalg
F → LF defines a homeomorphism Φ from Min(M(A)) onto
MinPrimal(M(A)). The method of proof of Theorem 3.2 is similar to that of [28,
Theorem 3.2] except that we are here working with filters of zero sets rather than
with ideals of cozero sets. For further work on the space of minimal (algebraic)
primal ideals of a C∗
-algebra, see [29] and [30].
For a C∗
-algebra B and a ∈ B, let Ia be the closed ideal of B generated
by a. The following lemma is a special case of [28, Theorem 2.3], which itself is a
special case of a more general result due to Keimel [18]. Recall that ideals are not
necessarily closed unless stated to be so.
Lemma 3.1. Let B be a C∗
-algebra and let P be a primal ideal of B. Then P is
a minimal primal ideal if and only if for all a ∈ P there exist b1, . . . , bn ∈ B P
such that IaIb1 . . . Ibn = {0}.
Let I⊥
a be the largest ideal of B such that IaI⊥
a = {0}. Then Lemma 3.1 implies
that if P is a minimal primal ideal of B and a ∈ P then I⊥⊥
a ⊆ P.](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-37-320.jpg)
![24 R.J. Archbold and D.W.B. Somerset
Theorem 3.2. Let A be a σ-unital quasi-standard C∗
-algebra and set X=Glimm(A).
Then the assignment P → Lalg
Z[P ] defines a homeomorphism Θ from Min(CR(X))
onto Min(M(A)).
Proof. First we show that if F = Z[P] for P ∈ Min(CR(X)) then Lalg
F is a minimal
primal ideal of M(A). Let bi ∈ M(A) Lalg
F (1 ≤ i ≤ n). Then Z(bi) /
∈ F for each
i, so Z(b1)∪· · ·∪Z(bn) /
∈ F since F is a prime z-filter. Hence Z(b1)∪· · ·∪Z(bn) =
X, so there exists x ∈ X such that bi /
∈ ˜
Jx (1 ≤ i ≤ n). Since ˜
Jx is primal,
b1M(A) . . . M(A)bn = {0}. Hence Lalg
F is primal. Now let b ∈ Lalg
F with b = 0. Then
Z(b) ∈ F, so by [19, Lemma 3.1] there exists f ∈ CR(X) such that Z(f)∪Z(b) = X
and Z(f) /
∈ F. Let c ∈ A with Z(c) = Z(f). Then Z(c) /
∈ F so c /
∈ Lalg
F , and
Z(c) ∪ Z(b) = X, so bM(A)c = {0} by Proposition 2.3(ii). This shows that Lalg
F is
a minimal primal ideal of M(A) and hence that Θ maps into Min(M(A)).
Now let P and Q be distinct elements of Min(CR(X)). Then Z[P] = Z[Q],
and since for each zero set Z there exists c ∈ A with Z(c) = Z, it follows that
Lalg
Z[P ] = Lalg
Z[Q]. This shows that Θ is injective.
Now suppose that Q ∈ Min(M(A)) and let G = {Z(b) : b ∈ Q}. We show
that G is a minimal prime z-filter on X. First note that if b ∈ Q then I⊥
b is
non-zero by Lemma 3.1, and indeed I⊥
b = {a ∈ M(A) : Z(a) ∪ Z(b) = X} by
the primality of the ideals ˜
Jx (x ∈ X). Hence Z(b) is non-empty, so ∅ /
∈ G.
For b, c ∈ Q, Z(b) ∩ Z(c) = Z(bb∗
+ cc∗
) ∈ G. If b ∈ Q and c ∈ M(A) with
Z(c) ⊇ Z(b) then Z(a) ∪ Z(c) = X for all a ∈ I⊥
b , so c ∈ I⊥⊥
b ⊆ Q, as observed
after Lemma 3.1. Hence Z(c) ∈ G. This shows that G is a proper z-filter, and also
that Q = Lalg
G . To show that G is a prime z-filter, let Z1 and Z2 be zero sets of X
such that Z1 ∪ Z2 = X. Let b, c ∈ A such that Z1 = Z(b) and Z2 = Z(c). Then
bM(A)c = {0}, so at least one of b and c (b say) belongs to Q since Q is primal.
Hence Z1 ∈ G. This shows that G is prime [14, 2E].
To see that G is minimal prime, let Z ∈ G and let b ∈ Q such that Z(b) = Z.
Then by Lemma 3.1 there exist c1, . . . , cn ∈ M(A)Q such that IbIc1 . . . Icn = {0}.
Hence Z(ci) /
∈ G (1 ≤ i ≤ n), by an argument in the previous paragraph, and
Z(b) ∪ Z(c1) ∪ · · · ∪ Z(cn) = X by the primality of the ideals ˜
Jx (x ∈ X). Set
Y = Z(c1) ∪ · · · ∪ Z(cn). Then Y is a zero set in X, being a finite union of zero
sets, and Y /
∈ G since G is prime. Since Z ∪Y = X it follows that no z-filter strictly
smaller than G can be prime. Hence G is a minimal prime z-filter, and Q = Lalg
G
belongs to the range of Θ. Thus Θ is a bijection.
Finally, for f ∈ CR(X) we can find a ∈ A such that Z(a) = Z(f); and
conversely, given a ∈ M(A), since A is σ-unital and quasi-standard we can find
f ∈ CR(X) such that Z(a) = Z(f). Hence in either case
Θ({P ∈ Min(CR(X)) : f /
∈ P}) = Θ({P ∈ Min(CR(X)) : Z(f) /
∈ Z[P]})
= {Lalg
Z[P ] ∈ Min(M(A)) : Z(a) /
∈ Z[P]}
= {Lalg
Z[P ] ∈ Min(M(A)) : a /
∈ Lalg
Z[P ]}.](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-38-320.jpg)
![Minimal Primal Ideals in the Multiplier Algebra 25
Since the hull-kernel topology on Min(CR(X)) can be defined either using ideals
or using elements, it follows that Θ is a homeomorphism.
A comparison of the proof of Theorem 3.2 with that of [28, Theorem 3.2]
shows that when A is a σ-unital quasi-standard C∗
-algebra, the assignment Q →
Q ∩ A gives a homeomorphism from Min(M(A)) onto Min(A).
For the next theorem, we need the following family of functions which is useful
for relating LF and Lalg
F . For 0 1/2, define the continuous piecewise linear
function f : [0, ∞) → [0, ∞) by: (i) f (x) = 0 (0 ≤ x ≤ ); (ii) f (x) = 2(x − )
( ≤ x ≤ 2 ); (iii) f (x) = x (2 ≤ x). Note that for b ∈ M(A)+
, if b ∈ LF then
f (b) belongs to the Pedersen ideal of LF for all [24, 5.6.1], and hence f (b) ∈ Lalg
F .
On the other hand, b − f (b) ≤ . Thus we have the following lemma.
Lemma 3.3. Let A be C0(X)-algebra with base map φ and let F be a z-filter on Xφ.
Let b ∈ M(A)+
. Then with the notation above, b ∈ LF if and only if f (b) ∈ Lalg
F
for all ∈ (0, 1/2).
Theorem 3.4. Let A be a σ-unital, quasi-standard C∗
-algebra with A/G non-unital
for all G ∈ Glimm(A) and set X = Glimm(A). Then the assignment P → LZ[P ]
defines a homeomorphism from Min(CR(X)) onto MinPrimal(M(A)).
Proof. By Theorem 3.2, it is enough to show that the assignment
Lalg
Z[P ] → LZ[P ] (P ∈ Min(CR(X)))
defines a homeomorphism Φ from Min(M(A)) onto MinPrimal(M(A)). If R is a
minimal closed primal ideal of M(A) then R contains some Lalg
Z[P ] ∈ Min(M(A)),
and hence R = LZ[P ]. Thus the range of Φ certainly contains MinPrimal(M(A)).
Furthermore, Theorem 2.4 implies that Φ is injective and also that if P, Q ∈
Min(CR(X)) with P = Q then LZ[P ] ⊆ LZ[Q]. Suppose that Q ∈ Min(CR(X)).
Then Lalg
Z[Q] ∈ Min(M(A)) so LZ[Q] is a closed primal ideal of M(A). Hence LZ[Q]
contains a minimal closed primal ideal of M(A), which we have just seen is of
the form LZ[P ] for P ∈ Min(CR(X)). Thus P = Q, so the range of Φ equals
MinPrimal(M(A)). Hence Φ is a bijection.
Now let a ∈ M(A)+
and let Z = Z(a), a zero set in X. Then by [7, Corollary
3.1] there exists c ∈ M(A)+
such that c + ˜
Jx = 1 for x ∈ X Z and c ∈ ˜
Jx for
x ∈ Z. Hence Z(f (c)) = Z for all ∈ (0, 1/2). Thus
Φ({Lalg
F ∈ Min(M(A)) : a /
∈ Lalg
F }) = Φ({Lalg
F ∈ Min(M(A)) : Z /
∈ F})
= {LF ∈ MinPrimal(M(A)) : c /
∈ LF },
by Lemma 3.3. On the other hand, again by Lemma 3.3,
Φ−1
({LF ∈ MinPrimal(M(A)) : a /
∈ LF })
=
∈(0,1/2)
{Lalg
F ∈ Min(M(A)) : f (a) /
∈ Lalg
F }).
Thus it follows that Φ is a homeomorphism.](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-39-320.jpg)
![26 R.J. Archbold and D.W.B. Somerset
Corollary 3.5. Let A be a σ-unital, continuous C0(X)-algebra with base map φ
such that A/Jx is non-unital and φ−1
(x) is contained in a limit set in Prim(A)
for all x ∈ Xφ. Then Min(CR(Xφ)) is homeomorphic to MinPrimal(M(A)).
Proof. By Lemma 2.1, A is quasi-standard and there is a homeomorphic map ψ :
Glimm(A) → Xφ. For G ∈ Glimm(A), there exists x ∈ Xφ such that ψ−1
(x) = G.
Hence Jx = G, so A/G is non-unital. The result now follows from Theorem 3.4.
4. Applications
The space of minimal prime ideals of CR(X) has been studied in numerous papers,
e.g., [19], [15], [12], [11], [16], so Theorem 3.4 has various immediate corollaries. We
present a sample of these. Recall that a topological space Y is countably compact
if every countable open cover of Y has a finite subcover. If Y is a T1-space then Y
is countably compact if and only if every infinite subset of Y has a limit point in
Y [23, p. 181].
Corollary 4.1. Let A be a σ-unital, quasi-standard C∗
-algebra with A/G non-unital
for all G ∈ Glimm(A).
(i) The Hausdorff space MinPrimal(M(A)) is totally disconnected and countably
compact.
(ii) If MinPrimal(M(A)) is locally compact then it is basically disconnected.
Proof. (i) The space of minimal closed primal ideals of a C∗
-algebra is always
Hausdorff in the τw-topology [2, Corollary 4.3]. The total disconnectedness and
countable compactness follow from Theorem 3.4 and from [15, Corollary 2.4] and
[15, Theorem 4.9] respectively.
(ii) This follows from Theorem 3.4 and [15, Theorem 4.7].
In the context of Corollary 4.1, recall that a necessary and sufficient condi-
tion for M(A) to be quasi-standard is that Glimm(M(A)) and MinPrimal(M(A))
should coincide both as sets and as topological spaces [5, Theorem 3.3]. Since
M(A) is unital, Glimm(M(A)) is compact, so MinPrimal(M(A)) would also have
to be compact. By Corollary 4.1(ii), this implies that MinPrimal(M(A)), and hence
Glimm(M(A)), would have to be basically disconnected; and this in turn implies
that Glimm(A) would have to be basically disconnected [14, 6M.1]. Thus we recover
the necessity of Glimm(A) being basically disconnected if M(A) is quasi-standard.
In point of fact, it was shown in [6, Corollary 4.9] that if A is a σ-unital quasi-
standard C∗
-algebra with centre equal to {0} then M(A) is quasi-standard if and
only if Glimm(A) is basically disconnected.
Corollary 4.2. Let A be a σ-unital, quasi-standard C∗
-algebra and suppose that
A/G is non-unital for all G ∈ Glimm(A). Then the following are equivalent:
(i) MinPrimal(M(A)) is compact;](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-40-320.jpg)
![Minimal Primal Ideals in the Multiplier Algebra 27
(ii) Glimm(A) is cozero-complemented; that is, for every cozero set U in
Glimm(A) there exists a cozero set V in Glimm(A) such that U ∩ V = ∅
and U ∪ V is dense in Glimm(A).
Proof. This follows by Theorem 3.4 and the characterization in [15, Corollary 5.5].
For example, if Glimm(A) is basically disconnected or is homeomorphic to an ordi-
nal space then Glimm(A) is cozero complemented [16, Examples 1.6], so the space
MinPrimal(M(A)) is compact. On the other hand, if Glimm(A) is the Alexandroff
double of a compact metric space without isolated points then Glimm(A) is com-
pact and first countable but not cozero complemented [16, Examples 1.7]. Hence
MinPrimal(M(A)) is not compact.
If A is separable, much more can be said. Recall that a regular closed set is
one that is the closure of its interior. If A is separable then Glimm(A) is perfectly
normal [8, Lemma 3.9] (i.e., every closed subset of Glimm(A) is a zero set) so A
certainly satisfies condition (ii) of the next corollary.
Corollary 4.3. Let A be a σ-unital, quasi-standard C∗
-algebra. Suppose that A/G
is non-unital for G ∈ Glimm(A). Then the following are equivalent:
(i) MinPrimal(M(A)) is compact and extremally disconnected;
(ii) Every regular closed set in Glimm(A) is the closure of a cozero set.
In particular, if A is separable then A satisfies these equivalent conditions.
Proof. This follows by Theorem 3.4 and the characterization in [15, Theorems 4.4
and 5.6].
More generally, recall that a topological space X has the countable chain con-
dition if every family of non-empty pairwise disjoint open subsets of X is countable.
It is easily seen that a completely regular topological space with the countable
chain condition has property (ii) of Corollary 4.3. If a C∗
-algebra A has a faithful
representation on a separable Hilbert space, then Glimm(A) satisfies the countable
chain condition [30, p. 85].
We conclude with one further application of Theorem 3.4.
Corollary 4.4. Set A = C(βN N) ⊗ K(H). Then MinPrimal(M(A)) is nowhere
locally compact. If Martin’s Axiom holds then MinPrimal(M(A)) is not an F-space.
Proof. Both statements follow from Theorem 3.4, the first by [15, Example 5.9],
and the second by [12, Corollary 4].](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-41-320.jpg)
![28 R.J. Archbold and D.W.B. Somerset
References
[1] C.A. Akemann, G.K. Pedersen and J. Tomiyama, Multipliers of C∗
-algebras. J.
Funct. Anal. 13 (1973), 277–301.
[2] R.J. Archbold, Topologies for primal ideals. J. London Math. Soc. (2) 36 (1987),
524–542.
[3] R.J. Archbold and C.J.K. Batty, On factorial states of operator algebras, III. J.
Operator Theory 15 (1986), 53–81.
[4] R.J. Archbold, E. Kaniuth and D.W.B. Somerset, Norms of inner derivations for
multipliers of C∗
-algebras and group C∗
-algebras. J. Functional Analysis 262 (2012),
2050–2073.
[5] R.J. Archbold and D.W.B. Somerset, Quasi-standard C∗
-algebras. Math. Proc.
Camb. Phil. Soc 107 (1990), 349–360.
[6] R.J. Archbold and D.W.B. Somerset, Multiplier algebras of C0(X)-algebras. Münster
J. Math. 4 (2011), 73–100.
[7] R.J. Archbold and D.W.B. Somerset, Ideals in the multiplier and corona algebras of
a C0(X)-algebra. J. London Math. Soc. (2) 85 (2012), 365–381.
[8] R.J. Archbold and D.W.B. Somerset, Spectral synthesis in the multiplier algebra of
a C0(X)-algebra. Quart. J. Math. Oxford 65 (2014), 1–24.
[9] R.J. Archbold and D.W.B. Somerset, Separation properties in the primitive ideal
space of a multiplier algebra. Israel J. Math. 200 (2014), 389–418.
[10] R.C. Busby, Double centralizers and extensions of C∗
-algebras. Trans. Amer. Math.
Soc. 132 (1968), 79–99.
[11] A. Dow, The space of minimal prime ideals of C(βN N) is probably not basically
disconnected, pp. 81–86, General Topology and Applications, (Lecture Notes in Pure
and Applied Math., 123), Dekker, NY, 1990.
[12] A. Dow, M. Henriksen, R. Kopperman and J. Vermeer, The space of minimal prime
ideals of C(X) need not be basically disconnected. Proc. Amer. Math. Soc. 104 (1988),
317–320.
[13] G.A. Elliott, Derivations of matroid C∗
-algebras, II. Ann. Math.(2) 100 (1974), 407–
422.
[14] L. Gillman and M. Jerison, Rings of Continuous Functions. Van Nostrand, New
Jersey, 1960.
[15] M. Henriksen and M. Jerison, The space of minimal prime ideals of a commutative
ring. Trans. Amer. Math. Soc. 115 (1965), 110–130.
[16] M. Henriksen and R.G. Woods, Cozero complemented space; when the space of min-
imal prime ideals of a C(X) is compact. Topology and Its Applications 141 (2004),
147–170.
[17] E. Kaniuth, G. Schlichting and K. Taylor, Minimal primal and Glimm ideal spaces
of group C∗
-algebras. J. Funct. Anal. 130 (1995), 45–76.
[18] K. Keimel, A unified theory of minimal prime ideals. Acta Math. Acad. Sci. Hun-
garicae 23 (1972), 51–69.
[19] J. Kist, Minimal prime ideals in commutative semigroups. Proc. London. Math. Soc.
(3) 13 (1963), 31–50.](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-42-320.jpg)
![Minimal Primal Ideals in the Multiplier Algebra 29
[20] D. Kucerovsky and P.W. Ng, Nonregular ideals in the multiplier algebra of a stable
C∗
-algebra. Houston J. Math. 33 (2007), 1117–1130.
[21] H.X. Lin, Ideals of multiplier algebras of simple AF-algebras. Proc. Amer. Math. Soc.
104 (1988), 239–244.
[22] M. Mandelker, Prime z-ideal structure of C(R). Fund. Math. 63 (1968), 145–166.
[23] J.R. Munkres, Topology. 2nd Edition, Prentice-Hall, New Jersey, 1999.
[24] G.K. Pedersen, C∗
-algebras and their Automorphism Groups. Academic Press, Lon-
don, 1979.
[25] F. Perera, Ideal structure of multiplier algebras of simple C∗
-algebras with real rank
zero. Can. J. Math. 53 (2001), 592–630.
[26] H.L. Pham, Uncountable families of prime z-ideals in C0(R). Bull. London Math.
Soc. 41 (2009), 354–366.
[27] M. Rørdam, Ideals in the multiplier algebra of a stable C∗
-algebra. J. Operator Th.
25 (1991), 283–298.
[28] D.W.B. Somerset, Minimal primal ideals in Banach algebras. Math. Proc. Camb.
Phil. Soc. 115 (1994), 39–52.
[29] D.W.B. Somerset, The local multiplier algebra of a C∗
-algebra. Quart. J. Math.
Oxford (2) 47 (1996), 123–132.
[30] D.W.B. Somerset, Minimal primal ideals in rings and Banach algebras. J. Pure and
Applied Algebra 144 (1999), 67–89.
[31] D.P. Williams, Crossed Products of C∗
-algebras. American Mathematical Society,
Rhode Island, 2007.
R.J. Archbold and D.W.B. Somerset
Institute of Mathematics
University of Aberdeen
Kings College
Aberdeen AB24 3UE
Scotland, UK
e-mail: r.archbold@abdn.ac.uk
dwbsomerset@gmail.com](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-43-320.jpg)
![Operator Theory:
Advances and Applications, Vol. 250, 31–48
Countable Spectrum, Transfinite Induction
and Stability
Wolfgang Arendt
Dedicated to Charles Batty on the occasion of his sixtieth birthday
Abstract. We reconsider the contour argument and proof by transfinite in-
duction of the ABLV-Theorem given in [AB88]. But here we use the method
to prove a Tauberian Theorem for Laplace transforms which has the ABVL-
Theorem about stability of a semigroup as corollary and also gives quantita-
tive estimates.
It is interesting that considering countable spectrum leads to the same
problems Cantor encountered when he tried to prove a uniqueness result for
trigonometric series. It led him to invent ordinal numbers and transfinite
induction. We explain these connections in the article.
Mathematics Subject Classification (2010). 47D06, 44A10.
Keywords. Semigroups, asymptotic behavior, Laplace transform, Tauberian
theorem, countable spectrum, transfinite induction, uniqueness theorem for
trigonometric series.
1. Introduction
Frequently, it is worthwhile to revisit a mathematical result with the benefit of
several years’ hindsight. Things may appear in a different light, different methods
might be known. The result I am talking about here is the stability theorem I
proved together with Charles Batty 25 years ago, which says the following. Let
(T (t))t≥0 be a bounded C0-semigroup with generator A. If σ(A) ∩ iR is countable
and σp(A
) ∩ iR = ∅ (where σp(A
) denotes the point spectrum of the adjoint),
then the semigroup is stable; i.e., limt→∞ T (t)x = 0 for all x ∈ X.
It was not necessary to wait 25 years for new methods to appear. In fact, ex-
actly at the same time, this stability result was obtained independently by Ljubich
and Vu [LV88] at the University of Kharkov in the Soviet Union by completely dif-
ferent methods. For this reason the result is frequently called the ABLV-Theorem.
Ljubich and Vu use a quotient method, and later Fourier methods were developed
by Esterle, Strouse and Zouakia [ESZ92] (spectral synthesis) and it was Chill
c
Springer International Publishing Switzerland 2015](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-44-320.jpg)
![32 W. Arendt
[Chi98] who shed new light onto the old methods of Ingham – 80 years after their
first appearance. These different approaches all have their advantages and mer-
its. The proof by Ljubich and Vu is the most functional analytical in nature. Its
disadvantage is that it merely works in the context of semigroups and not more
generally for Laplace transforms. The advantage of Chill’s approach is that it is
valid for Laplace transforms (see [Chi98], [ABHN11, Theorem 4.9.7]). We refer to
the survey by Chill and Tomilov [CT07] for more information and also to Section
5.5 of [ABHN11].
Still, we want to revisit our proof from 1986, which used two ingredients, a
contour argument and transfinite induction. Compared with the other methods,
there are two advantages: our proof is completely elementary and it also gives quan-
titative results (which have grown in importance recently, see Batty [Bat90], Batty
and Duykaerts [BD08], Borichev and Tomilov [BT10], Batty, Chill and Tomilov
[BCT], [BBT14] as well as Section 4.4 in [ABHN11]).
Concerning elegance and esthetics, the opinions of colleagues are not unan-
imous. Most people believe that our method is quite technical and even we did
not use it in our book [ABHN11] to prove the ABLV-Theorem. Still, we believe
that the transfinite induction argument we used in 1986 is quite striking and even
elegant. Once the inductive statement is formulated in the right way, its proof is
automatic. Our aim in this article is to make this transparent by formulating the
technical part in an abstract and easy way (Lemma 3.6). But we also arrange the
arguments differently and obtain a new interesting result, namely a (quantitative)
Tauberian theorem for Laplace transforms (Theorem 3.1) where an exceptional
countable set occurs in the hypothesis. It is this result which we prove by transfi-
nite induction in the present article (in contrast to our original proof [AB88] where
the argument by transfinite induction was done on the level of the semigroup). The
powerful Mittag-Leffler Theorem, a topological argument in the spirit of Baire’s
theorem, allows one to pass from the Tauberian theorem to the ABLV-Theorem,
see Section 4. Our Tauberian theorem gives also an improvement of a Tauberian
theorem for power series by Allan, O’Farrell and Randsford [AOR87] which was
motivated by the Katznelson–Tzafriri theorem.
Concerning the contour estimates, they demonstrate the power of Cauchy’s
Theorem and are most elegant when the spectrum on the imaginary axis is empty.
As an appetizer we consider this case in Section 2 emphasizing the quantitative
character. In Section 3 we prove the general Tauberian theorem elaborating the use
of transfinite induction. It is interesting that Cantor encountered similar problems
as we did in the context of countable spectrum when he tried to prove a uniqueness
result for trigonometric series where a closed, countable exceptional set has to be
mastered. It was this problem which led him to develop set theory, ordinal numbers
and transfinite induction. In Section 5 we take the opportunity to present the
solution of Cantor’s problem by transfinite induction, a striking resemblance to
our proof, a resemblance of which we were not aware in 1986.
Cantor must have been aware of the argument, but he never published the
end of the proof.](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-45-320.jpg)
![Countable Spectrum, Transfinite Induction and Stability 33
2. Empty spectrum
This section is an introduction to the subject where we consider the simplest case
of a complex Tauberian theorem, the Newman-trick for contour integrals and the
special case of the ABLV-Theorem where the spectrum on the imaginary axis is
empty. The results are contained in [AB88], [AP92] (see also [ABHN11]). Here
however, we put them together in a way which makes transparent the quantitative
nature of the results and which demonstrates the power of the contour argument
in a simple case. The more refined techniques are then presented in Section 3.
We consider a function f ∈ L∞
(R+, X) where X is a complex Banach space,
R+ = [0, ∞). By
ˆ
f(λ) :=
∞
0
e−λt
f(t) dt (Re λ 0)
we denote the Laplace transform of f. It is a holomorphic function defined on the
right half-plane C+.
If F(t) :=
t
0
f(s) ds converges to F∞ as t → ∞, then limλ→0
ˆ
f(λ) = F∞. This
Abelian theorem is easy to see. The converse is false in general: If limλ→∞
ˆ
f(λ) =
F∞ exists, then
t
0 f(s) ds need not converge as t → ∞. But if a theorem says
that it does under some additional hypothesis then we call it a Tauberian theorem
and the additional hypothesis a Tauberian condition. An interesting Tauberian
theorem is the following.
Theorem 2.1 (Newman–Korevaar–Zagier). Assume that ˆ
f has a holomorphic ex-
tension to an open set containing C+. Then
lim
t→∞
t
0
f(s) ds
exists.
It follows that limt→∞
t
0 f(s) ds = ˆ
f(0) by the remark above. Here the
Tauberian condition is that ˆ
f can be extended to a holomorphic function on an
open set containing C+. A theorem of this type had already been proved by Ing-
ham [Ing35] in the thirties (see also Korevaar’s book [Kor04, p. 135]). But Newman
[New80] found an elegant contour argument (which he applied to Dirichlet series),
that was used by Korevaar [Kor82] and Zagier [Zag97] for Laplace transforms to
give beautiful proofs of the prime number theorem. Here is an estimate, which
implies Theorem 2.1 and which shows the simplicity of the argument as well as its
quantitative aspect.
We let f∞ := supt≥0 f(t).
Proposition 2.2. Let R 0. Assume that ˆ
f has a holomorphic extension to a
neighborhood of C+ ∪ i[−R, R]. Then
lim sup
t→∞
t
0
f(s) − ˆ
f(0)
≤
f∞
R
.](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-46-320.jpg)
![34 W. Arendt
Proof. Let g = ˆ
f and for t 0 let
gt(z) =
t
0
e−zs
f(s) ds.
Thus gt is an entire function. Let U be an open, simply connected set containing
i[−R, R]∪C+. Denote by γ a path going from iR to −iR lying entirely in U ∩{z ∈
C : Re z 0} besides the endpoints.
We apply Cauchy’s Theorem to this contour. The introduction of an ad-
ditional fudge factor under the following integral is the ingenious trick due to
Newman.
t
0
f(s) ds − ˆ
f(0) = gt(0) − g(0)
=
1
2πi
|z|=R
Re z0
(gt(z) − g(z))etz
1 +
z2
R2
dz
z
+
1
2πi
γ
(gt(z) − g(z))etz
1 +
z2
R2
dz
z
=: I1(t) + I2(t).
It follows from the Dominated Convergence Theorem that limt→∞ I2(t) = 0.
In order to estimate I1(t) let z = Reiθ
, |θ| π
2 , be on the right-hand semi-
circle. Then on the one hand
(gt(z) − g(z))etz
=
∞
t
e−zs
f(s) ds etz
≤ f∞
∞
t
e−sR cos θ
ds etR cos θ
≤
f∞
R cos θ
and on the other
|1 +
z2
R2
| = |1 + ei2θ
| = |e−iθ
+ eiθ
|
= 2 cos θ.
Thus
I1(t) ≤
1
2π
π
f∞
R cos θ
2 cosθ =
f∞
R
and the proposition is proved.
In 1986 when we worked in Oxford on stability of semigroups we knew a
version of Theorem 2.1 from an unpublished manuscript by Zagier (cf. [Zag97]).
It was easy to apply it to semigroups:
Let (T (t))t≥0 be a C0-semigroup with generator A. Assume that T (t) ≤ M
for all t ≥ 0. For x ∈ X let f(t) = T (t)x. Then ˆ
f(λ) = R(λ, A)x. Now assume that](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-47-320.jpg)
![Countable Spectrum, Transfinite Induction and Stability 35
σ(A) ∩ iR = ∅. Then ˆ
f(0) = −A−1
x and the Newman–Korevaar–Zagier Theorem
2.1 implies that
t
0
f(s) ds =
t
0
T (s)AA−1
x ds
= T (t)A−1
x − A−1
x
converges to −A−1
x as t → ∞.
Hence T (t)A−1
x → 0 as t → ∞ for all x ∈ X. Since rg A−1
= D(A) is dense
in X and T (t) ≤ M it follows that limt→∞ T (t)x = 0 for all x ∈ X; i.e., the
semigroup is stable. We have proved the following.
Theorem 2.3. Assume that (T (t))t≥0 is a bounded C0-semigroup with generator A.
If σ(A) ∩ iR = ∅, then limt→∞ T (t)x = 0 for all x ∈ X.
It is natural to ask what happens if σ(A) ∩ iR = ∅. If iη ∈ σp(A
), the point
spectrum of the adjoint A
of A, then T (t)
x
= eiηt
x
for all t ≥ 0 and some
x
∈ X
{0}. Let x ∈ X be such that x
, x = 1. Then T (t)x, x
= eiηt
for all
t ≥ 0 and so the semigroup is definitely not stable. Thus
σp(A
) ∩ iR = ∅ (2.1)
is a necessary condition for stability.
By the Hahn–Banach Theorem, condition (2.1) is equivalent to
rg(iη − A) being dense in X (2.2)
where rg stands for the range of the operator. What is special for x ∈ rg(iη − A)?
Let x = (iη − A)y where y ∈ D(A), f(t) = T (t)x as before. Then
t
0
f(s)e−iηs
ds = y − e−iηt
T (t)y.
Thus
sup
t≥0
t
0
f(s)e−iηs
ds
∞. (2.3)
This condition turns out to be useful for proving a Tauberian theorem by the
contour method if iη is a singular point.
Before discussing this in the next section we point out a generalization of
the Tauberian Theorem 2.1. It is not necessary to assume that a holomorphic
extension exists, a continuous extension suffices.
Theorem 2.4. Let f ∈ L∞
(R+, X), R 0, F∞ ∈ X. Assume that 1
λ ( ˆ
f(λ) − F∞)
has a continuous extension to C+ ∪ i[−R, R]. Then
lim sup
t→∞
t
0
f(s) ds − F∞
≤
2f∞
R
. (2.4)](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-48-320.jpg)
![36 W. Arendt
This is obtained by a modification of the contour argument above (cf. [AP92,
Lemma 5.2], where a more complicated situation is considered). We give the proof
of Theorem 2.4 in order to be complete. It is interesting that now, instead of
the Dominated Convergence Theorem, we use the Riemann–Lebesgue Theorem
for Fourier coefficients. The price is a factor 2 appearing in the estimate (2.4) in
contrast to the better estimate given in Proposition 2.2.
Proof. First case: F∞ = 0.
Let g = ˆ
f. Thus g(z)
z has a continuous extension to C+ ∪ i[−R, R]. By (a
slight extension of) Cauchy’s Theorem one has
γ
g(z)
z
(1 +
z2
R2
)etz
dz +
|z|=R
Re z0
g(z)
z
(1 +
z2
R2
)etz
dz = 0 (2.5)
where γ is the straight line from iR to −iR.
For t 0 consider the entire function
gt(z) =
t
0
e−sz
f(s) ds.
Thus by (2.5),
t
0
f(s) ds =
1
2πi
|z|=R
gt(z) 1 +
z2
R2
etz dz
z
=
1
2πi
|z|=R
Re z0
(gt(z) − g(z)) 1 +
z2
R2
etz dz
z
−
1
2πi
γ
g(z)etz
1 +
z2
R2
dz
z
+
1
2πi
|z|=R
Re z0
gt(z) 1 +
z2
R2
etz dz
z
=: I1(t) + I2(t) + I3(t).
By the Riemann–Lebesgue Theorem, limt→∞ I2(t) = 0.
One has I1(t) ≤ 1
R f∞ for all t ≥ 0 as in Proposition 2.2.
The integral I3(t) can be estimated in a similar way,
lim sup
t→∞
I3(t) ≤
1
R
f∞.
Thus lim supt→∞
t
0 f(s) ds ≤ 2
R f∞.
Second case: F∞ ∈ X is arbitrary.
Let ϕ: [0,∞) → R be continuous with compact support satisfying
1
0
ϕ(s)ds = 1.
Let f1(t) := f(t) − ϕ(t)F∞. Then ˆ
f1(λ) = ˆ
f(λ) − ϕ̂(λ)F∞, ϕ̂(0) = 1.
Thus
ˆ
f1(λ)
λ
=
ˆ
f(λ) − F∞
λ
−
ϕ̂(λ) − ϕ̂(0)
λ
F∞
has a continuous extension to C+ ∪ i[−R, R].](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-49-320.jpg)
![Countable Spectrum, Transfinite Induction and Stability 37
By the first case
lim sup
t→∞
t
0
f(s) ds − F∞
≤
2
R
f∞.
Applying the preceding results to f(· + s) instead of f one even obtains the
estimate
lim sup
t→∞
t
0
f(s) ds − F∞
≤
1
R
lim sup
t→∞
f(t), (2.6)
in Proposition 2.2 and the estimate
lim sup
t→∞
t
0
f(s) ds − F∞
≤
2
R
lim sup
t→∞
f(t), (2.7)
instead of (2.4), cf. [AP92, Remark 9.2].
We finish this section by going back to the origins of Tauberian theory. Given
a bounded sequence (an)n∈N0 consider the power series p(z) =
∞
n=0 anzn
which
is defined for |z| 1. If
∞
n=0 an =: b∞ exists then Abel showed in 1826 that
lim
x 1
p(x) = b∞. (2.8)
The converse is not true in general. Additional assumptions are needed. It was
Tauber who proved in 1897 that the series converges if in addition to (2.8) one
assumes that
lim
n→∞
nan = 0, (2.9)
thus proving the first “Tauberian theorem”.
Littlewood showed in 1911 that the “Tauberian condition” (2.9) can be re-
laxed to supn∈N nan ∞.
Another Tauberian theorem is due to Riesz. It is actually a consequence of
the estimate (2.6).
Theorem 2.5 (Riesz). Let an ∈ X, n ∈ N0, such that limn→∞ an = 0. Assume
that the power series
p(z) =
∞
n=0
anzn
(|z| 1)
has a holomorphic extension to an open neighborhood of 1. Then
∞
n=0
an = p(1).
Proof. Let f(t) = an if t ∈ [n, n + 1). Then f ∈ L∞
(R+, X) and
ˆ
f(λ) =
1 − e−λ
λ
p(e−λ
) (Re λ 0).](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-50-320.jpg)
![38 W. Arendt
Thus ˆ
f has a holomorphic extension to a disc of radius 2R centered at 0 for some
R 0 and ˆ
f = p(1). Thus (2.6) implies that
lim
n→∞
n
k=0
ak − p(1)
≤
1
R
lim sup
n→∞
an = 0.
This proof is taken from [AP92, Remark 3.4].
3. A complex Tauberian theorem
Let f ∈ L∞
(R+, X). The Laplace transform ˆ
f of f is a holomorphic function from
the open right-hand half-plane C+ into X.
If F(t) :=
t
0
f(s) ds converges to F∞ as t → ∞, then limλ 0
ˆ
f(λ) = F∞ by
an easy Abelian theorem. As in Section 2, we want to prove the converse. But here
we will relax the assumptions considerably. As in Theorem 2.4 we will estimate
lim sup
t→∞
F(t) − F∞.
The Tauberian condition is expressed in terms of the boundary behavior of ˆ
f(λ)
as λ → iη.
Theorem 3.1. Let R 0, F∞ ∈ X. Let E ⊂ (−R, 0) ∪ (0, R) be closed and
countable. Assume that
(a)
ˆ
f(λ)−F∞
λ has a continuous extension to C+ ∪ i([−R, R] E) and that
(b) supt≥0
t
0
e−iηs
f(s) ds ∞ for all η ∈ E.
Then
lim sup
t→∞
t
0
f(s) ds − F∞
≤
2f∞
R
. (3.1)
Our point is that the bound in (b) may depend on η ∈ E. In the case where
it is independent, (3.1) can be proved purely by a contour argument (see [AP92,
Theorem 3.1], and [AB88, Theorem 4.1] for a slightly more special case). Since
we do not assume a uniform bound in (b) our proof needs an argument of trans-
finite induction. It is similar to the transfinite induction argument given for the
proof of the ABLV-Theorem in [AB88] and, we think, an interesting mathematical
argument in its own right. Here it is.
As in the proof of Proposition 2.4 we may assume that F∞ = 0 which we do
now. We assume the hypotheses of Theorem 3.1.
For the proof we denote by Jn the set of all (η1, . . . , ηn, 1, . . . , n) with ηj ∈
E, j 0 such that the intervals (ηj − j, ηj + j) are pairwise disjoint and 0 ∈
n
j=1[ηj − j, ηj + j] ⊂ (−R, R). Given a set K ⊂ (−R, 0) ∪ (0, R) we say that
(η1, . . . , ηn, 1, . . . , n) covers K, if K ⊂
n
j=1(ηj − j, ηj + j). With the help of
these notations the basic estimate can be formulated as follows.
Lemma 3.2 (basic estimate). There exist functions an, bn : Jn → (0, ∞) satisfying
for all n, p ∈ N](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-51-320.jpg)
![Countable Spectrum, Transfinite Induction and Stability 39
(a) an(η1, . . . , ηn, 1, . . . , n) → 1 as ( 1, . . . , n) → 0 in Rn
(b) an+p(η1, . . . , ηn+p, 1, . . . , n+p) → an(η1, . . . , ηn, 1, . . . , n)
as ( n+1, . . . , n+p) → 0 in Rp
(c) bn(η1, . . . , ηn, 1, . . . , n) → 0 as ( 1, . . . , n) → 0 in Rn
(d) bn+p(η1, . . . , ηn+p, 1, . . . , n+p) → bn(η1, . . . , ηn, 1, . . . , n)
as ( n+1, . . . , n+p) → 0 in Rp
such that the following holds:
If E is covered by (η1, . . . , ηn, 1, . . . , n) ∈ Jn then
lim sup
t→∞
t
0
f(s) ds
≤
2f∞
R
an(η1, . . . , ηn, 1, . . . , n) + bn(η1, . . . , ηn, 1, . . . , n).
These estimates are obtained by changing the contour in the proof of The-
orem 2.4 on the straight line i[−R, R] by introducing semicircles of radius j,
j = 1, . . . , n. For the proof we refer to [AP92, Lemma 5.2] (which is a modification
of [AB88, Lemma 3.1]).
Remark. The reader might better understand the proof of [AP92, Lemma 5.2] by
replacing “and 0 =” on line 11, 12 of p. 430 by a “−” and lifting the term to the
end of line 10. Also the signs “+” on lines 15 and 17 should be replaced by a “−”.
Proof of Theorem 3.1. Let E0 := E ∩[−R, R]. Thus E0 is compact and countable.
Given an ordinal α we define Eα inductively by
Eα =
⎧
⎪
⎪
⎨
⎪
⎪
⎩
the set of all cluster points
of Eα, if α is a successor ordinal;
βα
Eβ, if α is a limit ordinal.
We will prove that the following statement S(α) holds for all ordinals α:
S(α) : if Eα = ∅, then
lim sup
t→∞
t
0
f(s) ds
≤
2
R
f∞ (3.2)
and if Eα is covered by (η1, . . . , ηn, 1, . . . , n) ∈ Jn then
lim sup
t→∞
t
0
f(s) ds
≤
2f∞
R
an(η1, . . . , ηn, 1, . . . , n) + bn(η1, . . . , ηn, 1, . . . , n).
(3.3)
Once this statement is proved the proof of the theorem is completed as fol-
lows:
Since Eα is compact and countable it possesses an isolated point whenever
Eα is non empty. Thus Eα+1 Eα whenever Eα = ∅. This implies that Eα0 = ∅
for some α0 (see Proposition 5.2). Hence statement S(α0) gives the result.](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-52-320.jpg)
![40 W. Arendt
Now we prove that S(α) holds for all ordinals α.
α = 0: If E0 = ∅, this is Theorem 2.4. If E0 = ∅, then this follows immediately
from the basic estimate Lemma 3.2.
α 0: Assume that S(β) holds for all β α. We show that S(α) holds.
First case: α is a limit ordinal.
Then Eα = βα Eβ.
If Eα = ∅, then there exists β α such that Eβ = ∅. The inductive hypothesis
implies that (3.2) holds.
If Eα is covered by (η1, . . . , ηn, 1, . . . , n) ∈ Jn, then there exists β α such
that Eβ is covered by (η1, . . . , ηn, 1, . . . , n). Thus (3.3) follows by the inductive
hypothesis.
Second case: α is a successor ordinal.
If Eα = ∅, then Eα−1 is finite, say Eα−1 = {η1, . . . , ηn}. Choose j 0 so
small that (η1, . . . , ηn, 1, . . . , n) ∈ Jn. Then it follows by the inductive hypothesis
that
lim sup
t→∞
t
0
f(s) ds
≤
2f∞
R
an(η1, . . . , ηn, 1, . . . , n) + bn(η1, . . . , ηn, 1, . . . , n).
Letting ( 1, . . . , n) → 0 in Rn
yields (3.2).
If Eα is covered by (η1, . . . , ηn, 1, . . . , n) ∈ Jn, then
Eα−1
n
j=1
(ηj − j, ηj + j)
is finite, consisting of, say, {ηn+1, . . . , ηn+p}. Choose j 0, j = n+1, . . ., n+p, so
small that (η1, . . . , ηn+p, 1, . . . , n+p) ∈ Jn+p. Then by the inductive hypothesis
lim sup
t→∞
t
0
f(s) ds
≤
2f∞
R
an+p(η1, . . . , ηn+p, 1, . . . , n+p) + bn+p(η1, . . . , ηn+p, 1, . . . , n+p).
Sending ( n+1, . . . , n+p) to 0 in Rp
gives the desired estimate (3.3).
Thus S(α) is proved.
Remark 3.3. Applying Theorem 3.1 to the function f(·+s) instead of f one obtains
the estimate
lim sup
t→∞
t
0
f(s) ds − F∞ ≤
2
R
lim sup
t→∞
f(t) (3.4)
which improves (3.1), cf. [AP92, Remark 3.2].
The following is an immediate consequence of Theorem 3.1.](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-53-320.jpg)
![Countable Spectrum, Transfinite Induction and Stability 41
Corollary 3.4. Let E ⊂ R be closed and countable such that 0 ∈ E. Let F∞ ∈ X.
Assume that
(a)
ˆ
f(λ)−F∞
λ has a continuous extension to C+ iE and that
(b) supt≥0
t
0 e−iηs
f(s) ds ∞ for all η ∈ E.
Then limt→∞
t
0 f(s) ds = F∞.
Remark. In the case where (a) is replaced by the stronger hypothesis
(a
) ˆ
f has a holomorphic extension to an open set containing C+ iE,
Corollary 3.4 is proved by Batty, van Nerven and Räbiger [BvNR98, Theorem 4.3],
where a slightly weaker hypothesis than (b) is considered (cf. [BvNR98, Remark 2].
The methods are very different though.
We may transform Corollary 3.4 into a Tauberian theorem of a different type
where convergence of f(t) as t → ∞ is the conclusion.
Corollary 3.5. Let f ∈ L∞
(0, ∞; X) and f∞ ∈ X. Assume that
ˆ
f(λ) −
f∞
λ
(Re λ 0)
has a continuous extension to C+ iE where E ⊂ R is closed, countable and 0 ∈ E.
Assume that
sup
t≥0
t
0
e−iηs
f(s) ds
∞ for all η ∈ E. (3.5)
Then limt→∞
1
δ
δ+t
t
f(s) ds = f∞ for all δ 0.
If f is uniformly continuous on [τ, ∞) for some τ 0, then
lim
t→∞
f(t) = f∞.
This follows from Corollary 3.4 as [AP92, Theorem 3.5] follows from [AP92,
Theorem 3.1]. We refer to Chill [Chi98], [ABHN11, Theorem 4.9.7] for a different
approach via Fourier Analysis to such Tauberian theorems.
Finally, we apply Corollary 3.5 to power series. By D := {z ∈ C : |z| 1}
we denote the unit disc and by Γ := {z ∈ C : |z| = 1} the unit circle.
Corollary 3.6. Let an ∈ X, supn∈N0
an ∞, p(z) =
∞
n=0 anzn
for z ∈ D. Let
F be a closed, countable subset of Γ such that
(a) p has a continuous extension to D F and
(b) supN∈N
N
n=0 anzn
∞ for all z ∈ Γ.
Then limn→∞ an = 0
It follows from Riesz’ Theorem 2.5 that
∞
n=0
anzn
= p(z)
for all z ∈ D F.](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-54-320.jpg)
![This ebook is for the use of anyone anywhere in the United States
and most other parts of the world at no cost and with almost no
restrictions whatsoever. You may copy it, give it away or re-use it
under the terms of the Project Gutenberg License included with this
ebook or online at www.gutenberg.org. If you are not located in the
United States, you will have to check the laws of the country where
you are located before using this eBook.
Title: Faux's Memorable Days in America, 1819-20; and Welby's Visit
to North America, 1819-20, part 2 (1820)
Author: W. Faux
Adlard Welby
Editor: Reuben Gold Thwaites
Release date: March 18, 2013 [eBook #42364]
Most recently updated: October 23, 2024
Language: English
Credits: Produced by Greg Bergquist and the Online Distributed
Proofreading Team at http://www.pgdp.net (This file was
produced from images generously made available by The
Internet Archive/American Libraries.)
*** START OF THE PROJECT GUTENBERG EBOOK FAUX'S
MEMORABLE DAYS IN AMERICA, 1819-20; AND WELBY'S VISIT TO
NORTH AMERICA, 1819-20, PART 2 (1820) ***](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-60-320.jpg)
![JOURNAL
(PART II)
January 1st, 1820.—I left Princeton at ten o'clock, with Mr. Phillips
and Mr. Wheeler; and here parted with my good and kind friend
Ingle.
I met and spoke, ten miles off, with two hog-jobbing judges,
Judge Prince and Judge Daniel,104 driving home twenty fat hogs,
which they had just bought.
I reached, and rested at Petersburgh,105 consisting of fifteen
houses. I passed good farms. Our landlord of this infant town,
though having an [333] ostler, was compelled to groom, saddle,
unsaddle, and to do all himself. Having fifty dollars owing to him,
from a gentleman of Evansville, he arrested him, when he went into
the bounds; then he sued one of the bondsmen, who also entered
the bounds. The squire is next to be sued, who, it is expected, will
do likewise.
Sunday, 2nd.—I rode thirty-one miles this day, and rested at
Edmonstone, in a little cold log-hole, out of which I turned an
officer's black cat, which jumped from the roof into our faces, while
in bed; but she soon found her way in again, through a hole in the
roof. The cat liked our fire. We got no coffee nor tea, but cold milk
and pork, and corn cake.
3rd.—Travelled all day, through the mud-holes formed by springs
running from countless hills, covered with fine timber, to breakfast,
at three o'clock, p. m. I supped and slept at Judge Chambers's, a
comfortable house, and saw again the judge's mother, of eighty,
whose activity and superior horsemanship, I have before mentioned.
I smoked a segar with Mrs. Judge, while she smoked her pipe, (the
first pipe I have seen here.) She, as well as the old lady, is a quaker.](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-67-320.jpg)
![The judge was gone to the metropolitan town of Coridon, being a
senator, on duty.106 The land which I passed over all this day,
seemed poor, but full of wild turkeys and bears.
4th.—I reached Miller's to supper, but found no [334] coffee; cold
milk only, as a substitute. The ride hither is interesting, through a
fine rolling country. The wolves howled around us all night.
5th.—Passed the Silver Hills,107 from the summit of which is a
fine, extensive prospect of Kentucky, the Ohio, and of Louisville,
where we breakfasted. I called with Mr. Flower's letter to Archer,
who was out. I received the present of a cow-hide whip, from a lady,
and promised to treat the beast kindly, for her sake. Judge
Waggoner recently shook hands at a whiskey-shop, with a man
coming before him that day, to be tried for murder. He drank his
health, and wished him well through.
I rode seven miles with an intelligent old Kentucky planter, having
four children, who cultivate his farm, without negroes. He says,
Kentucky is morally and physically ruined. We have been brought
up to live without labour: all are demoralized. No man's word or
judgment is to be taken for the guidance and government of
another. Deception is a trade, and all are rogues. The west has the
scum of all the earth. Long ago it was said, when a man left other
States, he is gone to hell, or Kentucky. The people are none the
better for a free, good government. The oldest first settlers are all
gone or ruined. Your colt, sir, of one hundred dollars, is worth only
fifteen dollars. At Louisville, as good a horse can be bought at ten
dollars, or fifteen dollars. You are therefore cheated.
The Missouri territory boasts the best land in [335] the country,
but is not watered by springs. Wells are, however, dug, abounding in
good water, says our hearty landlord, just returned from viewing that
country.
The bottom land is the finest in the world. Corn, from sixty to
eighty bushels, and wheat, from forty to sixty bushels an acre. The
best prairies are full of fine grass, flowers, and weeds, not coarse,
benty, sticky grass, which denotes the worst of prairie land. Grass, of](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-68-320.jpg)
![a short fine quality, fit for pasture or hay, every where abounds. The
country is full of wild honey, some houses having made seven and
eight barrels this season, taken out of the trees, which are cut down
without killing the bees. These industrious insects do not sting, but
are easily hived and made tame. Our landlord likes the Missouri, but
not so well as Old Kentucky.
Two grim, gaunt-looking men burst into our room, at two, this
morning; and by six, the landlord disturbed us by cow-hiding his
negro, threatening to squeeze the life out of him.
6th.—I rode all day through a country of fine plantations, and
reached Frankfort to supper, with the legislative body, where I again
met my gay fellow-traveller, Mr. Cowen. It was interesting to look
down our table, and contemplate the many bright, intelligent faces
around me: men who might honour any nation. As strangers, we
were [336] invited by the landlord, (the best I have seen) to the first
rush for a chance at the table's head.
7th.—I travelled this day through a fine country of rich pasture
and tillage, to Lexington City, to Keen's excellent tavern. I drank
wine with Mr. Lidiard, who is removing eastward, having spent
1,100l. in living, and travelling to and fro. Fine beef at three cents
per lb. Fat fowls, one dollar per dozen. Who would not live in old
Kentucky's first city?
8th.—Being a wet day, I rested all day and this night. Prairie flies
bleed horses nearly to death. Smoke and fire is a refuge to these
distressed animals. The Indian summer smoke reaches to the Isle of
Madeira.
Visited the Athenæum. Viewed some fine horses, at two hundred
dollars each.
Sunday, 9th.—I quitted Lexington, and one of the best taverns in
America, for Paris, Kentucky, and a good, genteel farm-house, the
General Washington, twenty-three miles from the city, belonging to
Mr. Hit, who, though owning between four hundred and five hundred
acres of the finest land in Kentucky, does not think it beneath him to
entertain travellers and their horses, on the best fare and beds in](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-69-320.jpg)
![the country. He has been offered sixty dollars, and could now have
forty dollars an acre, for his land, which averages thirty bushels of
wheat, and sixty bushels of corn per acre, and, in [337] natural or
artificial grass, is the first in the world. Sheep, (fine stores) one
dollar per head; beef, fine, three cents per lb., and fowls, one dollar
per dozen.
10th.—Rode all day in the rain and mud, and through the worst
roads in the universe, frequently crossing creeks, belly deep of our
horses. Passed the creek at Blue-lick, belly deep, with sulphurous
water running from a sulphur spring, once a salt spring. The water
stinks like the putrid stagnant water of an English horse-pond, full of
animal dung. This is resorted to for health.
Five or six dirkings and stabbings took place, this fall, in Kentucky.
11th.—Breakfasted at Washington, (Kentucky) where we parted
with Mr. Phillips, and met the Squire, and another gentleman,
debating about law. Rested at Maysville, a good house, having
chambers, and good beds, with curtains. The steam-boats pass this
handsome river town, at the rate of fifteen to twenty miles an hour.
To the passenger, the effect is beautiful, every minute presenting
new objects of attraction.
12th.—Crossed the Ohio in a flat, submitting to Kentuckyan
imposition of seventy-five cents a horse, instead of twenty-five,
because we were supposed to be Yankees. We will not, said the
boat-man, take you over, for less than a dollar each. We heard of
you, yesterday. The gentleman in the cap (meaning me) looks as
though he [338] could afford to pay, and besides, he is so slick with
his tongue. The Yankees are the smartest of fellows, except the
Kentuckyans. Sauciness and impudence are characteristic of these
boat-men, who wished I would commence a bridge over the river.
Reached Union town, Ohio,108 and rested for the night.
13th.—Breakfasted at Colonel Wood's. A fine breakfast on beef,
pork-steaks, eggs, and coffee, and plenty for our horses, all for fifty
cents each. Slept at Colonel Peril's, an old Virginian revolutionary
soldier, living on 400 acres of fine land, in a good house, on an](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-70-320.jpg)
![eminence, which he has held two years only. He now wishes to sell
all at ten dollars an acre, less than it cost him, because he has a
family who will all want as much land each, in the Missouri, at two
dollars. He never had a negro. He knows us to be English from our
dialect. We passed, this day, through two or three young villages.
14th.—Breakfasted at Bainbridge,109 where is good bottom land,
at twenty to thirty dollars an acre, with improvements. The old
Virginian complains of want of labourers. A farmer must do all
himself. Received of our landlady a lump of Ohio wild sugar, of which
some families make from six to ten barrels a-year, sweet and good
enough.
Reached Chilicothé, on the Sciota river, to [339] sup and rest at
the tavern of Mr. Madera, a sensible young man. Here I met Mr.
Randolph, a gentleman of Philadelphia, from Missouri and Illinois,
who thinks both sickly, and not to be preferred to the east, or others
parts of the west. I saw three or four good houses, in the best
street, abandoned, and the windows and doors rotting out for want
of occupants.
15th.—I rode all day through a fine interesting country, abounding
with every good thing, and full of springs and streams. Near
Lancaster,110 I passed a large high ridge of rocks, which nature has
clothed in everlasting green, being beautified with the spruce,
waving like feathers, on their bleak, barren tops. I reached Lancaster
to rest; a handsome county seat, near which land is selling
occasionally from sixteen dollars to twenty dollars. A fine farm of
170 acres, 100 being cleared, with all improvements, was sold lately
by the sheriff, at sixteen dollars one cent an acre, much less than it
cost. Labour is to be had at fifty cents and board, but as the produce
is so low, it is thought farming, by hired hands, does not pay. Wheat,
fifty cents; corn, 33½ cents; potatoes, 33 cents a bushel; beef, four
dollars per cwt.; pork, three dollars; mutton none; sheep being kept
only for the wool, and bought in common at 2s. 8d. per head.
Met Judge and General ——, who states that four millions of acres
of land will this year [340] be offered to sale, bordering on the lakes.](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-71-320.jpg)
![Why then should people go to the Missouri? It is not healthy near
the lakes, on account of stagnant waters, made by sand bars, at the
mouth of lake rivers. The regular periodical rising and falling of the
lakes is not yet accounted for. There is no sensible diminution, or
increase of the lake-waters. A grand canal is to be completed in five
years, when boats will travel.111
Sunday, 16th.—I left Lancaster at peep of day, travelling through
intense cold and icy roads to Somerset, eighteen miles, in five hours,
to breakfast.112 Warmed at an old quarter-section man, a Dutch
American, from Pennsylvania. He came here eleven years since,
cleared seventy acres, has eight children, likes his land, but says,
produce is too low to make it worth raising. People comfortably
settled in the east, on good farms, should stay, unless their children
can come and work on the land. He and his young family do all the
work. Has a fine stove below, warming the first, and all other floors,
by a pipe passing through them.
I slept at a good tavern, the keeper of which is a farmer. All are
farmers, and all the best farmers are tavern-keepers. Farms,
therefore, on the road, sell from 50 to 100 per cent more than land
lying back, though it is no better in quality, and for mere farming,
worth no more. But on the road, a farm and frequented tavern is
found to be [341] a very beneficial mode of using land; the produce
selling for double and treble what it will bring at market, and also
fetching ready money. Labour is not to be commanded, says our
landlord.
17th.—Started at peep of day in a snow-storm, which had covered
the ground six inches deep. Breakfasted at beautiful Zanesville, a
town most delightfully situated amongst the hills. Twelve miles from
this town, one Chandler, in boring for salt, hit upon silver; a mine,
seven feet thick, 150 feet below the surface. It is very pure ore, and
the proprietor has given up two acres of the land to persons who
have applied to the legislature to be incorporated. He is to receive
one-fifth of the net profits.](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-72-320.jpg)
![18th.—I rode all day through a fine hilly country, full of springs
and fountains. The land is more adapted for good pasture than for
cultivation. Our landlord, Mr. Gill, states that wheat at fifty cents is
too low; but, even at that price, there is no market, nor at any other.
In some former years, Orleans was a market, but now it gets
supplied from countries more conveniently situated than Ohio, from
which it costs one dollar, or one dollar and a quarter per barrel, to
send it. Boats carrying from 100 to 500 barrels, sell for only 16
dollars.
From a conversation, with an intelligent High Sheriff of this county,
I learn that no common debtor has ever lain in prison longer than
five [342] days. None need be longer in giving security for the
surrender of all property.
19th.—Reached Wheeling late at night, passing through a
romantic, broken, mountainous country, with many fine springs and
creeks. Thus I left Ohio, which, thirty years ago, was a frontier state,
full of Indians, without a white man's house, between Wheeling,
Kaskasky, and St. Louis.
20th.—Reached Washington, Pennsylvania, to sleep, and found
our tavern full of thirsty classics, from the seminary in this town.
21st.—Reached Pittsburgh, through a beautiful country of hills, fit
only for pasture. I viewed the fine covered bridges over the two
rivers Monongahela and Allegany, which cost 10,000 dollars each.
The hills around the city shut it in, and make the descent into it
frightfully precipitous. It is most eligibly situated amidst rocks, or
rather hills, of coal, stone, and iron, the coals lying up to the
surface, ready for use. One of these hills, or coal banks, has been
long on fire, and resembles a volcano. Bountiful nature has done
every thing for this rising Birmingham of America.
We slept at Wheeling, at the good hotel of Major Spriggs, one of
General Washington's revolutionary officers, now near 80, a
chronicle of years departed.113
22nd.—Bought a fine buffalo robe for five dollars. [343] The
buffaloes, when Kentucky was first settled, were shot, by the](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-73-320.jpg)
![settlers, merely for their tongues; the carcase and skin being
thought worth nothing, were left where the animal fell.
Left Pittsburgh for Greensburgh, travelling through a fine,
cultivated, thickly settled country, full of neat, flourishing, and good
farms, the occupants of which are said to be rich. Land, on the road,
is worth from fifteen to thirty dollars; from it, five to fifteen dollars
per acre. The hills and mountains seem full of coal-mines and stone-
quarries, or rather banks of coal and stone ever open gratuitously to
all. The people about here are economical and intelligent; qualities
characteristic of Pennsylvania.
Sunday, 23d.—We agreed to rest here until the morrow; finding
one of our best horses sick; and went to Pittsburgh church.
24th.—My fellow traveller finding his horse getting worse, gave
him away for our tavern bill of two days, thus paying 175 dollars for
two days board. While this fine animal remained ours, no doctor
could be found, but as soon as he became our landlord's, one was
discovered, who engaged to cure him in a week. Mr. Wheeler took
my horse, and left me to come on in the stage, to meet again at
Chambersburgh.
The country round about here is fine, but there is no market,
except at Baltimore, at five dollars a barrel for flour. The carriage
costs two and half [344] dollars. I saw two young ladies, Dutch
farmers' daughters, smoking segars in our tavern, very freely, and
made one of their party. Paid twelve dollars for fare to
Chambersburgh.114
Invited to a sleying party of ten gentlemen, one of whom was the
venerable speaker (Brady) of the senate of this state. They were
nearly all drunk with apple-toddy, a large bowl of which was handed
to every drinker. One gentleman returned with a cracked skull.
25th.—Left this town, at three o'clock in the morning, in the stage,
and met again at Bedford, and parted, perhaps, for ever, with my
agreeable fellow-traveller, Mr. Wheeler, who passed on to New York.
Passed the Laurel-hill, a huge mountain, covered with everlasting](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-74-320.jpg)
![green, and a refuge for bears, one of which was recently killed with
a pig of 150lbs. weight in his mouth.
26th.—Again mounted my horse, passing the lonely Allegany
mountains, all day, in a blinding snow-storm, rendering the air as
dense as a November fog in London. Previous to its coming on, I
found my naked nose in danger. The noses of others were wrapped
up in flannel bags, or cots, and masks for the eyes, which are liable
to freeze into balls of ice.
Passed several flourishing villages. The people here seem more
economical and simple, than in other states. Rested at M'Connell's
town, 100 miles from Washington city.
[345] 27th.—Crossed the last of the huge Allegany mountains,
called the North Mount, nine miles over, and very high. My horse
was belly deep in snow.
Breakfasted at Mercersburgh, at the foot of the above mountain,
and at the commencement of that fine and richest valley in the
eastern states, in which Hagar's town stands, and which extends
through Pennsylvania, Maryland, and Virginia, from 100 to 200 miles
long, and from 30 to 40 broad. Land here, three years ago, sold at
100 to 120 dollars, although now at a forced sale, 160 acres sold for
only 1,600 dollars, with improvements, in Pennsylvania. And if, says
my informant, the state makes no law to prevent it, much must
come into the market, without money to buy, except at a ruinous
depreciation.
Passed Hagar's town, to Boonsburgh, to rest all night, after 37
miles travel.
The old Pennsylvanian farmer, in answer to How do you do
without negroes? said, Better than with them. I occupy of my
father 80 acres in this valley, and hire all my hands, and sell five
loads of flour, while some of the Marylanders and Virginians cannot
raise enough to maintain their negroes, who do but little work.
28th.—Breakfasted on the road; passed Middletown, with two fine
spires, a good town; and also Frederick town, a noble inland town,
and next to Lancaster, in Pennsylvania, and the first [346] in the](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-75-320.jpg)
![United States. It has three beautiful spires. It is much like a second
rate English town, but not so cleanly; something is dirty, or in ruins.
It stands at the foot of the Blue Ridge, in the finest, largest vale in
the world, running from the eastern sea to the Gulf of Mexico.
Rested at Windmiller's, a stage-house, thirty miles from
Washington, distinguished only by infamous, ungenerous, extortion
from travellers. Here I paid 75 cents for tea; 25 cents for a pint of
beer, 9s. sterling for a bushel of oats and corn, and 50 cents for hay
for the night. The horse cost 6s. 9d. in one night.
29th.—Rode from seven till eleven o'clock, sixteen miles to
breakfast, at Montgomery-court-house, all drenched in rain. I
reached Washington city, at six this evening. Here, for the first time,
I met friend Joseph Lancaster, full of visionary schemes, which are
unlikely to produce him bread.
Sunday, 30th.—Went to Congress-hall, and heard grave senators
wrangling about slavery. Governor Barbour spoke with eloquence.
Friend Lancaster's daily and familiar calls on the great, and on his
Excellency, the President, about schooling the Indians, and his
praises of the members, are likely to wear out all his former fame,
already much in ruins. I was this day introduced by him to —— Parr,
Esq., an English gentleman of fortune, from Boston, Lincolnshire,
[347] who has just returned from a pedestrian pilgrimage to
Birkbeck and the western country.
February 1st.—I again went to Congress, where I heard Mr.
Randolph's good speech on the Missouri question. This sensible
orator continually refers to English authors and orators, insomuch
that all seemed English. These American statesmen cannot open
their mouths without acknowledging their British origin and
obligations.—I shall here insert some observations on the
constitution and laws of this country, and on several of the most
distinguished members of Congress, for which I am indebted to the
pen of G. Waterstone, Esq., Congressional Librarian at
Washington.115](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-76-320.jpg)
![Observations on the Constitution and Laws of the United States, with
Sketches of some of the most prominent public Characters.
Like the Minerva of the ancients, the American people have
sprung, at once, into full and vigorous maturity, without the
imbecility of infancy, or the tedious process of gradual progression.
They possess none of the thoughtless liberality and inconsiderate
confidence of youth; but are, already, distinguished by the cold and
cautious policy of declining life, rendered suspicious by a long
acquaintance with the deceptions and the vices of the world.
Practitioners of jurisprudence have become [348] almost
innumerable, and the great end of all laws, the security and
protection of the citizen, is in some degree defeated. It is to the
multiplicity and ambiguity of the laws of his age, that Tacitus has
ascribed most of the miseries which were then experienced; and this
evil will always be felt where they are ambiguous and too numerous.
In vain do the Americans urge that their laws have been founded on
those of England, the wisdom and excellence of which have been so
highly and extravagantly eulogized. The difference, as Mably
correctly observes, between the situation of this country and that is
prodigious;116 the government of one having been formed in an age
of refinement and civilization, and that of the other, amidst the
darkness and barbarism of feudal ignorance. In most of the states
the civil and criminal code is defective; and the latter, like that of
Draco, is often written in blood. Why should not each state form a
code of laws for itself, and cast off this slavish dependence on Great
Britain, whom they pretend so much to dislike?
With a view of explaining more perfectly the nature of this
constitution, I will briefly exhibit the points in which the British and
American governments differ [349].
In England. In America.
I. The king
possesses imperial
dignity.
There is no king;
the president acts as](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-77-320.jpg)
![the weights and
measures, and can
alone coin money and
give currency to
foreign coin.
IX. He is the
universal proprietor of
the kingdom.
The president has
nothing to do with the
property of the United
States [350].
X. The king's
person is sacred and
inviolate; he is
accountable to no
human power, and
can do no wrong.
The president is
nothing more than an
individual, is
amenable like all civil
officers, and
considered as capable
of doing wrong as any
other citizen.
XI. The British
legislature contains a
house of lords, 300
nobles, whose seats,
honours, and
privileges are
hereditary.
There are no
nobles, and both
houses of Congress
are elected.
It may, perhaps, be unnecessary to adduce more points of
difference to illustrate the nature of the American government.
These are amply sufficient to demonstrate the entire democratic
tendency of the constitution of the United States, and the error
under which those persons labour, who believe that but few
differences, and those immaterial and unimportant, exist between
these two governments. They have, indeed, in common the Habeas
Corpus and the Trial by Jury, the great bulwarks of civil liberty, but in
almost every other particular they disagree.
The second branch of this government is the legislature. This
consists of a Senate and House of Representatives; the members of](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-79-320.jpg)
![the latter are chosen every two years by the people; and those of
the former, every six years by the legislatures of the different states.
It is in this branch that the American government differs from the
republics of ancient and modern times; it is this which [351] makes
it not a pure, but a representative democracy; and it is this which
gives it such a decided superiority over all the governments in the
world. Experience has demonstrated the impracticability of
assembling a numerous collection of people to frame laws, and their
incompetency, when assembled, for judicious deliberation and
prompt and unbiassed decision. The passions of illiterate and
unthinking men are easily roused into action and inflamed to
madness. Artful and designing demagogues are too apt to take
advantage of those imbecilities of our nature, and to convert them to
the basest purposes.
The qualifications of representatives are very simple. It is only
required that they should be citizens of the United States, and have
attained the age of twenty-five. The moment their period of service
expires, they are again, unless re-elected, reduced to the rank and
condition of citizens. If they should have acted in opposition to the
wishes and interests of their constituents, while performing the
functions of legislation, the people possess the remedy and can
exercise it without endangering the peace and harmony of society;
the offending member is dropped, and his place supplied by another,
more worthy of confidence. This consciousness of responsibility, on
the part of the representatives, operates as a perpetual guarantee to
the people, and protects and secures them in the enjoyment of their
political and civil liberties.
[352] It must be admitted that the Americans have attained the
Ultima Thulé in representative legislation, and that they enjoy this
inestimable blessing to a much greater extent than the people of
Great Britain. Of the three distinct and independent branches of that
government, one only owes its existence to the free suffrages of the
people, and this, from the inequality of representation, the long
intervals between the periods of election, and the liability of
members, from this circumstance, to be corrupted, is not so](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-80-320.jpg)
![important and useful a branch as might otherwise be expected.
Imperfect, however, as it is, the people, without it, would indeed be
slaves, and the government nothing more than a pure monarchy.
The American walks abroad in the majesty of freedom; if he be
innocent, he shrinks not from the gaze of upstart and insignificant
wealth, nor sinks beneath the oppression of his fellow-man.
Conscious of his rights and of the security he enjoys, by the liberal
institutions of his country, independence beams in his eye, and
humanity glows in his heart. Has he done wrong? He knows the
limits of his punishment, and the character of his judges. Is he
innocent? He feels that no power on earth can crush him. What a
condition is this, compared with that of the subjects of almost all the
European nations!
As long as it is preserved, the security of the citizen and the union
of the states, will be guaranteed, [353] and the country thus
governed, will become the home of the free, the retreat of misery,
and the asylum of persecuted humanity. As a written compact, it is a
phenomenon in politics, an unprecedented and perfect example of
representative democracy, to which the attention of mankind is now
enthusiastically directed. Most happily and exquisitely organized, the
American constitution is, in truth, at once a monument of genius,
and an edifice of strength and majesty. The union of its parts forms
its solidity, and the harmony of its proportions constitutes its beauty.
May it always be preserved inviolate by the gallant and highminded
people of America, and may they never forget that its destruction
will be the inevitable death-blow of liberty, and the probable
passport to universal despotism!
The speaker of the House of Representatives is Mr. Clay, a
delegate from Kentucky, and who, not long ago, acted a conspicuous
part, as one of the American commissioners at Ghent.117 He is a tall,
thin, and not very muscular man; his gait is stately, but swinging;
and his countenance, while it indicates genius, denotes dissipation.
As an orator, Mr. Clay stands high in the estimation of his
countrymen, but he does not possess much gracefulness or elegance
of manner; his eloquence is impetuous and vehement; it rolls like a](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-81-320.jpg)
![torrent, but like a torrent which is sometimes irregular, and
occasionally obstructed. Though there is a [354] want of rapidity and
fluency in his elocution, yet he has a great deal of fire and vigour in
his expression. When he speaks he is full of animation and
earnestness; his face brightens, his eye beams with additional lustre,
and his whole figure indicates that he is entirely occupied with the
subject on which his eloquence is employed. In action, on which
Demosthenes laid such peculiar emphasis, and which was so highly
esteemed among the ancients, Mr. Clay is neither very graceful nor
very imposing. He does not, in the language of Shakespear, so suit
the word to the action, and the action to the word, as not to o'erstep
the modesty of nature. In his gesticulation and attitudes, there is
sometimes an uniformity and awkwardness that lessen his merit as
an orator, and in some measure destroy the impression and effect
his eloquence would otherwise produce. Mr. Clay does not seem to
have studied rhetoric as a science, or to have paid much attention to
those artificial divisions and rhetorical graces and ornaments on
which the orators of antiquity so strongly insist. Indeed, oratory as
an art is but little studied in this country. Public speakers here trust
almost entirely to the efficacy of their own native powers for success
in the different fields of eloquence, and search not for the extrinsic
embellishments and facilities of art. It is but rarely they unite the
Attic and Rhodian manner, and still more rarely do they devote their
attention to the acquisition [355] of those accomplishments which
were, in the refined ages of Greece and Rome, considered so
essential to the completion of an orator. Mr. Clay, however, is an
eloquent speaker; and notwithstanding the defects I have
mentioned, very seldom fails to please and convince. His mind is so
organized that he overcomes the difficulties of abstruse and
complicated subjects, apparently without the toil of investigation or
the labour of profound research. It is rich, and active, and rapid,
grasping at one glance, connections the most distant, and
consequences the most remote, and breaking down the trammels of
error and the cobwebs of sophistry. When he rises to speak he
always commands attention, and almost always satisfies the mind on
which his eloquence is intended to operate. The warmth and fervor](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-82-320.jpg)
![of his feelings, and the natural impetuosity of his character, which
seem to be common to the Kentuckians, often indeed lead him to
the adoption of opinions, which are not, at all times, consistent with
the dictates of sound policy. Though ambitious and persevering, his
intentions are good and his heart is pure; he is propelled by a love of
country, but yet is solicitous of distinction; he wishes to attain the
pinnacle of greatness without infringing the liberties, or marring the
prosperity of that land of which it seems to be his glory to be a
native.
[356] The prominent traits of Mr. Clay's mind are quickness,
penetration, and acuteness; a fertile invention, discriminating
judgment, and good memory. His attention does not seem to have
been much devoted to literary or scientific pursuits, unconnected
with his profession; but fertile in resources, and abounding in
expedients, he is seldom at a loss, and if he is not at all times able
to amplify and embellish, he but rarely fails to do justice to the
subject which has called forth his eloquence. On the most
complicated questions, his observations made immediately and on
the spur of the occasion, are generally such as would be suggested
by long and deep reflection. In short, Mr. Clay has been gifted by
nature with great intellectual superiority, which will always give him
a decided influence in whatever sphere it may be his destiny to
revolve.
Mr. Clay's manners are plain and easy. He has nothing in him of
that reserve which checks confidence, and which some politicians
assume; his views of mankind are enlarged and liberal; and his
conduct as a politician and a statesman has been marked with the
same enlarged and liberal policy. As Speaker of the House of
Representatives, he presides generally with great dignity, and
decides on questions of order, sometimes, indeed, with too much
precipitation, but almost always correctly. It is but seldom his
decisions are disputed, [357] and when they are, they are not often
reversed.118
A Statesman, says Mirabeau, presents to the mind the idea of a
vast genius improved by experience, capable of embracing the mass](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-83-320.jpg)
![of social interests, and of perceiving how to maintain true harmony
among the individuals of which society is composed, and an extent
of information which may give substance and union to the different
operations of government.
Mr. Pinkney119 is between fifty and sixty years of age; his form is
sufficiently elevated and compact to be graceful, and his
countenance, though marked by the lines of dissipation, and rather
too heavy, is not unprepossessing or repulsive. His eye is rapid in its
motion, and beams with the animation of genius; but his lips are too
thick, and his cheeks too fleshy and loose for beauty; there is too a
degree of foppery, and sometimes of splendor, manifested in the
decoration of his person, which is not perfectly reconcileable to our
[358] ideas of mental superiority, and an appearance of
voluptuousness about him which cannot surely be a source of pride
or of gratification to one whose mind is so capacious and elegant. It
is not improbable, however, that this character is assumed merely for
the purpose of exciting a higher admiration of his powers, by
inducing a belief that, without the labour of study or the toil of
investigation, he can attain the object of his wishes and become
eminent, without deigning to resort to that painful drudgery by
which meaner minds and inferior intellects are enabled to arrive at
excellence and distinction. At the first glance, you would imagine Mr.
Pinkney was one of those butterflies of fashion, a dandy, known by
their extravagant eccentricities of dress, and peculiarities of
manners; and no one could believe, from his external appearance,
that he was, in the least degree, intellectually superior to his fellow
men. But Mr. Pinkney is indeed a wonderful man, and one of those
beings whom the lover of human nature feels a delight in
contemplating. His mind is of the very first order; quick, expanded,
fervid, and powerful. The hearer is at a loss which most to admire,
the vigour of his judgment, the fertility of his invention, the strength
of his memory, or the power of his imagination. Each of these
faculties he possesses in an equal degree of perfection, and each is
displayed in its full maturity, when the [359] magnitude of the
subject on which he descants renders its operation necessary. This](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-84-320.jpg)
![singular union of the rare and precious gifts of nature, has received
all the strength which education could afford, and all the polish and
splendour which art could bestow. Under the cloak of dissipation and
voluptuousness his application has been indefatigable, and his
studies unintermitted: the oil of the midnight lamp has been
exhausted, and the labyrinths of knowledge have been explored.
Mr. Pinkney is never unprepared, and never off his guard. He
encounters his subject with a mind rich in all the gifts of nature, and
fraught with all the resources of art and study. He enters the list with
his antagonist, armed, like the ancient cavalier, cap-a-pee; and is
alike prepared to wield the lance, or to handle the sword, as
occasion may require. In cases which embrace all the complications
and intricacies of law, where reason seems to be lost in the chaos of
technical perplexity, and obscurity and darkness assume the
dignified character of science, he displays an extent of research, a
range of investigation, a lucidness of reasoning, and a fervor and
brilliancy of thought, that excite our wonder, and elicit our
admiration. On the driest, most abstract, and uninteresting questions
of law, when no mind can anticipate such an occurrence, he
occasionally blazes forth in all the enchanting exuberance of a
chastened, but rich [360] and vivid imagination, and paints in a
manner as classical as it is splendid, and as polished as it is brilliant.
In the higher grades of eloquence, where the passions and feelings
of our nature are roused to action, or lulled to tranquillity, Mr.
Pinkney is still the great magician, whose power is resistless, and
whose touch is fascination. His eloquence becomes sublime and
impassioned, majestic and overwhelming. In calmer moments, when
these passions are hushed, and more tempered feelings have
assumed the place of agitation and disorder, he weaves around you
the fairy circles of fancy, and calls up the golden palaces and
magnificent scenes of enchantment. You listen with rapture as he
rolls along: his defects vanish, and you are not conscious of any
thing but what he pleases to infuse. From his tongue, like that of
Nestor, language more sweet than honey flows; and the attention
is constantly rivetted by the successive operation of the different](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-85-320.jpg)
![faculties of the mind. There are no awkward pauses, no hesitation
for want of words or of arguments: he moves forward with a pace
sometimes majestic, sometimes graceful, but always captivating and
elegant. His order is lucid, his reasoning logical, his diction select,
magnificent, and appropriate, and his style, flowing, oratorical, and
beautiful. The most laboured and finished composition could not be
better than that which he seems to utter [361] spontaneously and
without effort. His judgment, invention, memory, and imagination, all
conspire to furnish him at once with whatever he may require to
enforce, embellish, or illustrate his subject. On the dullest topic he is
never dry: and no one leaves him without feeling an admiration of
his powers, that borders on enthusiasm. His satire is keen, but
delicate, and his wit is scintillating and brilliant. His treasure is
exhaustless, possessing the most extensive and varied information.
He never feels at a loss; and he ornaments and illustrates every
subject he touches. Nihil quod tetigit, non ornavit. He is never the
same; he uses no common place artifice to excite a momentary thrill
of admiration. He is not obliged to patch up and embellish a few
ordinary thoughts, or set off a few meagre and uninteresting facts.
His resources seem to be as unlimited as those of nature; and fresh
powers, and new beauties are exhibited, whenever his eloquence is
employed. A singular copiousness and felicity of thought and
expression, united to a magnificence of amplification, and a purity
and chastity of ornament, give to his eloquence a sort of
enchantment which it is difficult to describe.
Mr. Pinkney's mind is in a high degree poetical; it sometimes
wantons in the luxuriance of its own creations; but these creations
never violate the purity of classical taste and elegance. He [362]
loves to paint when there is no occasion to reason; and addresses
the imagination and passions, when the judgment has been satisfied
and enlightened. I speak of Mr. Pinkney at present as a forensic
orator. His career was too short to afford an opportunity of judging
of his parliamentary eloquence; and, perhaps, like Curran, he might
have failed in a field in which it was anticipated he would excel, or,
at least, retain his usual pre-eminence. Mr. Pinkney, I think, bears a](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-86-320.jpg)
![stronger resemblance to Burke than to Pitt; but, in some particulars,
he unites the excellences of both. He has the fancy and erudition of
the former, and the point, rapidity, and elocution of the latter.
Compared with his countrymen, he wants the vigour and striking
majesty of Clay, the originality and ingenuity of Calhoun; but, as a
rhetorician, he surpasses both. In his action, Mr. Pinkney has,
unfortunately, acquired a manner, borrowed, no doubt, from some
illustrious model, which is eminently uncouth and inelegant. It
consists in raising one leg on a bench or chair before him, and in
thrusting his right arm in a horizontal line from his side to its full
length in front. This action is uniform, and never varies or changes in
the most tranquil flow of sentiment, or the grandest burst of
impassioned eloquence. His voice, though not naturally good, has
been disciplined to modulation by art; and, if it is not always
musical, it is [363] never very harsh or offensive. Such is Mr. Pinkney
as an orator; as a diplomatist but little can be said that will add to
his reputation. In his official notes there is too much flippancy, and
too great diffuseness, for beauty or elegance of composition. It is
but seldom that the orator possesses the requisites of the writer;
and the fame which is acquired by the tongue sometimes evaporates
through the pen. As a writer he is inferior to the present Attorney-
General,120 who unites the powers of both in a high degree, and
thus in his own person illustrates the position which he has laid
down, as to the universality of genius.
Mr. R. King is a senator from the State of New York, and was
formerly the resident minister at the court of St. James's.121 He is
now about sixty years of age, above the middle size, and somewhat
inclined to corpulency. His countenance, when serious and
thoughtful, possesses a great deal of austerity and rigour; but at
other moments it is marked with placidity and benevolence. Among
his friends he is facetious and easy; but when with strangers,
reserved and distant; apparently indisposed to conversation, and
inclined to taciturnity; but when called out, his colloquial powers are
of no ordinary character, and his conversation becomes peculiarly
instructive, fascinating, and humourous. Mr. King has read and](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-87-320.jpg)
![reflected much; and though long in public life, his attention [364]
has not been exclusively devoted to the political sciences; for his
information on other subjects is equally matured and extensive. His
resources are numerous and multiplied, and can easily be called into
operation. In his parliamentary addresses he always displays a deep
and intimate knowledge of the subject under discussion, and never
fails to edify and instruct if he ceases to delight. He has read history
to become a statesman, and not for the mere gratification it affords.
He applies the experience of ages, which the historical muse
exhibits, to the general purposes of government, and thus reduces
to practice the mass of knowledge with which his mind is fraught
and embellished. As a legislator he is, perhaps, inferior to no man in
this country. The faculty of close and accurate observation by which
he is distinguished has enabled him to remark and treasure up every
fact of political importance, that has occurred since the organization
of the American government; and the citizen, as well as the stranger,
is often surprised at the minuteness of his historical details, and the
facility with which they are applied. With the various subjects
immediately connected with politics, he has made himself well
acquainted; and such is the strength of his memory, and the extent
of his information, that the accuracy of his statements is never
disputed. Mr. King, however, is somewhat of an [365] enthusiast,
and his feelings sometimes propel him to do that which his judgment
cannot sanction. When parties existed in this country, he belonged
to, and was considered to be the leader of what was denominated
the federal phalanx; and he has often, perhaps, been induced, from
the influence of party feeling, and the violence of party animosity, to
countenance measures that must have wounded his moral
sensibilities; and that now, when reason is suffered to dictate,
cannot but be deeply regretted. From a rapid survey of his political
and parliamentary career, it would appear that the fury of party has
betrayed him into the expression of sentiments, and the support of
measures, that were, in their character, revolting to his feelings; but
whatever he may have been charged with, his intentions, at least,
were pure, and his exertions, as he conceived, calculated for the
public good. He was indeed cried down by a class of emigrants from](https://image.slidesharecdn.com/2676842-250727010656-5c0cee86/85/Operator-Semigroups-Meet-Complex-Analysis-Harmonic-Analysis-And-Mathematical-Physics-1st-Edition-Wolfgang-Arendt-88-320.jpg)
Operator Semigroups Meet Complex Analysis Harmonic Analysis And Mathematical Physics 1st Edition Wolfgang Arendt
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- 5. Operator Theory Advances and Applications 250 Wolfgang Arendt Ralph Chill Yuri Tomilov Editors Operator Semigroups Meet Complex Analysis, Harmonic Analysis and Mathematical Physics
- 7. Operator Theory: Advances and Applications Founded in 1979 by Israel Gohberg Volume 250 Joseph A. Ball (Blacksburg, VA, USA) Harry Dym (Rehovot, Israel) Marinus A. Kaashoek (Amsterdam, The Netherlands) Heinz Langer (Wien, Austria) Christiane Tretter (Bern, Switzerland) Vadim Adamyan (Odessa, Ukraine) Albrecht Böttcher (Chemnitz, Germany) B. Malcolm Brown (Cardiff, UK) Raul Curto (Iowa, IA, USA) Fritz Gesztesy (Columbia, MO, USA) Pavel Kurasov (Stockholm, Sweden) Lewis A. Coburn (Buffalo, NY, USA) Ciprian Foias (College Station, TX, USA) J.William Helton (San Diego, CA, USA) Thomas Kailath (Stanford, CA, USA) Peter Lancaster (Calgary, Canada) Peter D. Lax (New York, NY, USA) Donald Sarason (Berkeley, CA, USA) Bernd Silbermann (Chemnitz, Germany) Harold Widom (Santa Cruz, CA, USA) Associate Editors: Honorary and Advisory Editorial Board: Editors: Wolfgang Arendt (Ulm, Germany) Vern Paulsen (Houston, TX, USA) Mihai Putinar (Santa Barbara, CA, USA) Ilya M. Spitkovsky (Williamsburg, VA, USA) Subseries Linear Operators and Linear Systems Subseries editors: Daniel Alpay (Beer Sheva, Israel) Birgit Jacob (Wuppertal, Germany) André C.M. Ran (Amsterdam, The Netherlands) Subseries Advances in Partial Differential Equations Subseries editors: Bert-Wolfgang Schulze (Potsdam, Germany) Michael Demuth (Clausthal, Germany) Jerome A. Goldstein (Memphis, TN, USA) Nobuyuki Tose (Yokohama, Japan) Ingo Witt (Göttingen, Germany)
- 8. Operator Semigroups Editors Wolfgang Arendt • Ralph Chill • Yuri Tomilov Meet Complex Analysis, Harmonic Analysis and Mathematical Physics
- 9. ISSN 0255-0156 ISSN 2296-4878 (electronic) Operator Theory: Advances and Applications ISBN 978-3-319-1 - ISBN 978-3-319-1 - (eBook) DOI 10.1007/978-3-319-1 - Library of Congress Control Number: Mathematics Subject Classification (2010): 30, 35, 42, 46, 47 Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.birkhauser-science.com) Editors 8493 8494 8494 Wolfgang Arendt Institute of Applied Analysis Ulm University Ulm, Germany Institute of Mathematics Yuri Tomilov Polish Academy of Sciences Warsaw, Poland 7 4 4 Ralph Chill TU Dresden Dresden, Germany Institute of Analysis 2015958022
- 10. Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix E.M. Ait Benhassi, S. Boulite, L. Maniar and R. Schnaubelt Polynomial Internal and External Stability of Well-posed Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 R.J. Archbold and D.W.B. Somerset Minimal Primal Ideals in the Multiplier Algebra of a C0(X)-algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 W. Arendt Countable Spectrum, Transfinite Induction and Stability . . . . . . . . . . . . 31 C.J.K. Batty, R. Chill and S. Srivastava Maximal Regularity in Interpolation Spaces for Second-order Cauchy Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 B.V. Rajarama Bhat and S. Srivastava Stability of Quantum Dynamical Semigroups . . . . . . . . . . . . . . . . . . . . . . . . 67 A. Bobrowski Families of Operators Describing Diffusion Through Permeable Membranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 D. Borisov, I. Nakić, C. Rose, M. Tautenhahn and I. Veselić Multiscale Unique Continuation Properties of Eigenfunctions . . . . . . . . 107 I. Chalendar, J. Esterle and J.R. Partington Dichotomy Results for Norm Estimates in Operator Semigroups . . . . . 119 T. Duyckaerts Estimates on Non-uniform Stability for Bounded Semigroups . . . . . . . . 133 A.F.M. ter Elst and E.M. Ouhabaz Convergence of the Dirichlet-to-Neumann Operator on Varying Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
- 11. vi Contents J. Esterle and E. Fašangová A Banach Algebra Approach to the Weak Spectral Mapping Theorem for Locally Compact Abelian Groups . . . . . . . . . . . . 155 St. Fackler Regularity Properties of Sectorial Operators: Counterexamples and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 D. Fang, B. Han and M. Hieber Global Existence Results for the Navier–Stokes Equations in the Rotational Framework in Fourier–Besov Spaces . . . . . . . . . . . . . . . 199 F. Gesztesy, Y. Latushkin, F. Sukochev and Yu. Tomilov Some Operator Bounds Employing Complex Interpolation Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 T.A. Gillespie Power-bounded Invertible Operators and Invertible Isometries on Lp Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 A. Gomilko and Yu. Tomilov Generation of Subordinated Holomorphic Semigroups via Yosida’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 T.P. Hytönen A Quantitative Coulhon–Lamberton Theorem . . . . . . . . . . . . . . . . . . . . . . . 273 L. Khadkhuu, D. Tsedenbayar and J. Zemánek An Analytic Family of Contractions Generated by the Volterra Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 L.J. Konrad Lattice Dilations of Bistochastic Semigroups . . . . . . . . . . . . . . . . . . . . . . . . 287 I. Lasiecka and R. Triggiani Domains of Fractional Powers of Matrix-valued Operators: A General Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 L. Molnár General Mazur–Ulam Type Theorems and Some Applications . . . . . . . 311 S. Monniaux Traces of Non-regular Vector Fields on Lipschitz Domains . . . . . . . . . . . 343 J. van Neerven The Lp -Poincaré Inequality for Analytic Ornstein–Uhlenbeck Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
- 12. Contents vii C.-K. Ng and N.-C. Wong A Murray–von Neumann Type Classification of C∗ -algebras . . . . . . . . . 369 R. Picard, S. Trostorff and M. Waurick Well-posedness via Monotonicity – an Overview . . . . . . . . . . . . . . . . . . . . . 397 J. Prüss Perturbations of Exponential Dichotomies for Hyperbolic Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 D.W. Robinson Gaussian and non-Gaussian Behaviour of Diffusion Processes . . . . . . . . 463 F.L. Schwenninger and H. Zwart Functional Calculus for C0-semigroups Using Infinite-dimensional Systems Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 J. Voigt On Self-adjoint Extensions of Symmetric Operators . . . . . . . . . . . . . . . . . 491
- 13. Professor Charles Batty together with the editors of this volume, Jurata 2010 (from left to right: R. Chill, C. Batty, W. Arendt, Y. Tomilov)
- 14. Preface The last fifteen years opened a new era for semigroup theory with the emphasis on applications of abstract results, often unexpected and often far away from tradi- tional ones. The aim of the conference held in Herrnhut in June 2013 was to bring together prominent experts around modern semigroup theory, harmonic analysis, complex analysis and mathematical physics, and to show a lively interplay be- tween all of those areas and even beyond them. In addition, the meeting honoured the sixtieth anniversary of Prof C.J.K. Batty, whose scientific achievements are an impressive illustration of the conference goal. The present conference proceedings provide an opportunity to see the power of abstract methods and techniques dealing successfully with a number of ap- plications stemming from classical analysis and mathematical physics. The sam- ple of diverse topics treated by the proceedings include partial differential equa- tions, martingale and Hilbert transforms, Banach and von Neumann algebras, Schrödinger operators, maximal regularity and Fourier multipliers, interpolation, operator-theoretical problems (concerning generation, perturbation and dilation, for example), and various qualitative and quantitative Tauberian theorems with an accent on transfinite induction and magics of Cantor. The organizers express their sincere gratitude to Volkswagenstiftung for their generous support of the Herrnhut conference and to Thomas Hempfling of Birk- häuser for the enjoyable cooperation. Ulm, Dresden and Warsaw, December 2014 Wolfgang Arendt, Ralph Chill, Yuri Tomilov
- 15. Operator Theory: Advances and Applications, Vol. 250, 1–16 Polynomial Internal and External Stability of Well-posed Linear Systems El Mustapha Ait Benhassi, Said Boulite, Lahcen Maniar and Roland Schnaubelt Abstract. We introduce polynomial stabilizability and detectability of well- posed systems in the sense that a feedback produces a polynomially stable C0-semigroup. Using these concepts, the polynomial stability of the given C0-semigroup governing the state equation can be characterized via polyno- mial bounds on the transfer function. We further give sufficient conditions for polynomial stabilizability and detectability in terms of decompositions into a polynomial stable and an observable part. Our approach relies on a recent characterization of polynomially stable C0-semigroups on a Hilbert space by resolvent estimates. Mathematics Subject Classification (2010). Primary: 93D25. Secondary: 47A55, 47D06, 93C25, 93D15. Keywords. Internal and external stability, polynomial stability, transfer func- tion, stabilizability, detectability, well-posed systems. 1. Introduction Weakly damped or weakly coupled linear wave type equations often have polyno- mially decaying classical solutions without being exponentially stable, see, e.g., [1], [2], [4], [5], [8], [15], [16], [17], [18], [23], and the references therein. In these con- tributions various methods have been used, partly based on resolvent estimates. Recently this spectral theory has been completed for the case of bounded semi- groups T (·) in a Hilbert space with generator A. Here one can now characterize the ‘polynomial stability’ T (t)(I − A)−1 ≤ ct−1/α , t ≥ 1, of T (·) by the polynomial bound R(iτ, A) ≤ c|τ|α , |τ| ≥ 1, on the resolvent of A. These results are due to Borichev and Tomilov in [7] and to Batty and Duyckaerts in [6], see also [5], [15] and [17] for earlier contributions. We describe this theory in the next section. In a This work is part of a cooperation project supported by DFG (Germany) and CNRST (Morocco). c Springer International Publishing Switzerland 2015
- 16. 2 E.M. Ait Benhassi, S. Boulite, L. Maniar and R. Schnaubelt polynomial stable system the spectrum of the generator may approach the imagi- nary axis as Im λ → ±∞. This already indicates that this concept is more subtle than exponential stability. For instance, so far robustness results for polynomial stability are restricted to small regularizing perturbations, see [19]. At least for bounded semigroups in a Hilbert space one has now a solid background which can be used in other areas such as control theory. In the context of observability this was already done in [11] (based on [5] at that time). In this paper we start an investigation of polynomial stabilizability and detectability. Stabilizability is one of the basic concepts and topics of linear systems theory. Let the state system be governed by a generator A on the state Hilbert space X, and let Y and U be the observation and the control Hilbert spaces, respectively. For a moment, we simply consider bounded control and observation operators and feedbacks. For a bounded control operator B : U −→ X we obtain the system x (t) = Ax(t) + Bu(t), t ≥ 0, x(0) = x0, (1.1) with the control u ∈ L2 loc(R+, U), the initial state x0 ∈ X and the state x(t) ∈ X at time t ≥ 0. This system is exponentially stabilizable if one can find a (bounded) feedback F : X −→ U such that the C0-semigroup TBF (·) solving the closed-loop system x (t) = Ax(t) + BFx(t), t ≥ 0, x(0) = x0, (1.2) is exponentially stable. Observe that A + BF generates TBF (·). For the dual concept of exponential detectability, one starts with a generator A and a bounded observation operator C : X −→ Y . The output of this system is y = CT (·)x0. One then looks for a (bounded) feedback H : Y −→ X such that the C0-semigroup THC(·) generated by A + HC becomes exponentially stable. In our paper we allow for unbounded observation operators C defined on D(A) and control operators B mapping into the larger space X−1 = D(A∗ )∗ , where the domains are equipped with the respective graph norm. Here one has to assume that the output map x0 → y and the input map u → x(t) are continuous. Such systems are called admissible, see the next section for a precise definition and further information. The monograph [24] investigates these notions in detail. In this framework one can in particular treat boundary control and observation of partial differential equations. In order to use the full system (A, B, C), one also has to assume the bound- edness of the input-output map u → y. This leads to the concept of a well-posed system, which was introduced by G. Weiss and others, see Section 2, the recent survey [25], and, e.g., [22], [27], [28]. In well-posed systems, the Laplace transform of the input-output map gives the transfer function of the system, which plays an important role in the present paper. For well-posed systems, it becomes more difficult to determine the generators of the feedback systems, cf. [28]. However, in our arguments we can avoid to use a precise description of these operators. For well-posed systems exponential stabilizability and detectability was discussed in many papers, see, e.g., [9], [12], [13], [20], [21], [29], and the references therein.
- 17. Polynomial Internal and External Stability 3 In this paper we will weaken the exponential stability of the feedback sys- tem in the above concepts to polynomial stability. Here the feedback systems are described by equations for the resolvents of the generators of given and the feed- back semigroup which are coupled via a perturbation term involving the feedback, see Definitions 3.1 and 3.1. In the study of the resulting concepts of polynomial stabilizability and detectability we pursue two main questions, also treated in the above papers. We show that a system possesses these properties if it can be decomposed into a polynomial stable and an observable part, see Theorem 4.6 and 4.7. In the exponential case, such results are often called pole-assignment if the stable part has a finite-dimensional complement. Actually one can derive exponential stabi- lizability from much weaker concepts (optimizability or the finite cost condition), see [9] or [29]. So far it is not clear whether such implications hold for the natural analogues of these concepts to the polynomial setting. Moreover, it is known that optimizability can be characterized by decompositions as above if the resolvent set of the generator contains a strip around iR, see [12] or [21]. In the polynomial setting one here has to fight against the fact that the spectrum may approach the imaginary axis at infinity. So far we only have partial results in this context, not treated below. The main part of our results is devoted to the relationship between poly- nomial stability of the given semigroup and polynomial estimates on the trans- fer function of the system. It is known that A generates an exponentially stable semigroup if (and only if) the system (A, B, C) is exponential stabilizable and detectable and its transfer function is bounded on the right half-plane, see [20] and also [29] for an extension to the concepts of optimizability and estimatibil- ity. (Note that the ‘only if’ implication is easily shown with 0 feedbacks.) The boundedness of the transfer function is called external stability. In Theorem 4.3 we extend these results to our setting, thus requiring polynomial stabilizability and detectability and that the transfer function grows at most polynomially as | Im λ| → ∞. (The latter condition may be called polynomial external stability.) If the involved semigroups are bounded, we then obtain polynomial stability of the order one expects, i.e., the sum of the orders in the assumption. The proofs are based on various estimates and manipulations of formulas connecting resolvents, the transfer functions and their variants. We further use the results polynomial stability from [6] and [7] mentioned above. If the given semigroup is not known to be bounded, then the available the- ory on polynomial stability does not give the above-indicated convergence order. However, in applications one can often check the boundedness of a semigroup by the dissipativity of its generator, possibly for an equivalent norm. Similarly one can characterize well-posed systems with energy dissipation (so-called scattering passive systems), see, e.g., [22]. Besides the given semigroup, here also the trans- fer function is contractive which leads to an improvement of our main result for scattering passive systems, see Corollary 4.4. In general, not much is known on the preservation of boundedness under perturbations. In Theorem 5 of the recent
- 18. 4 E.M. Ait Benhassi, S. Boulite, L. Maniar and R. Schnaubelt paper [19] one finds a result which requires smallness of the perturbations as maps into spaces between D(A) and X. In Proposition 4.5 we show the boundedness in the framework of the present paper. Our approach is based on a characterization of bounded semigroups in terms of L2 -norms of the resolvents of A and A∗ due to [10], see Proposition 2.4. In the next section we discuss the background on polynomial stability and well-posed systems. In Section 3 we introduce polynomial stabilizability and de- tectability and establish several basic estimates. The last section contains our main results on external polynomial stability and on sufficient criteria for polynomial stabilizability and detectability. 2. Polynomial stability and well-posed systems We first discuss polynomially stable semigroups. Throughout T (·) denotes a C0- semigroup on a Banach space X with generator A. There are numbers ∈ R and M ≥ 1 such that T (t) ≤ Met for all t ≥ 0. The infimum of these numbers is denoted by ω0(A). The semigroup is called bounded if T (t) ≤ M for all t ≥ 0. We fix some ω ω0(A). It is well known that then the fractional powers (ω − A)β exist for β ∈ R. They are bounded operators for β ≤ 0 and closed ones for β 0. The domain Xβ of (ω − A)β for β 0 is endowed with the norm given by xβ = (ω − A)β x. The fractional powers satisfy the power law and coincide with usual powers for β ∈ Z. In particular, (ω − A)−β is the inverse of (ω − A)β for all β ∈ R. We next recall a definition from [5]. Definition 2.1. A C0-semigroup T (·) is called polynomially stable (of order α 0) if there is a constant α 0 such that T (t)(ω − A)−α ≤ ct−1 for all t ≥ 1. (Here and below, we write c 0 for a generic constant.) Note that a larger order α means a weaker convergence property. Due to Proposition 3.1 of [5], a bounded C0-semigroup T (·) is polynomially stable of order α 0 if and only if T (t)(ω − A)−αγ ≤ c(γ) t−γ , t ≥ 1, (2.1) for all/some γ 0. (There is also a partial extension to general C0-semigroups.) Combined with (2.1), Proposition 3 of [6] yields the following necessary con- dition for polynomial stability of bounded C0-semigroups. Here we set C± = {λ ∈ C Re λ ≷ 0} and Cr = r + C+ for r ∈ R. Proposition 2.2. Let T (·) be a bounded C0-semigroup which is polynomially stable of order α 0. Then the spectrum σ(A) of A belongs to C− and its resolvent is bounded by R(λ, A) ≤ c (1 + |λ|)α for all λ ∈ C+. (2.2)
- 19. Polynomial Internal and External Stability 5 Due to Lemma 3.2 in [14], the estimate (2.2) is true if and only if R(λ, A)(ω − A)−α ≤ c for all λ ∈ C+. (2.3) If one drops the boundedness assumption, the above result still holds with an epsilon loss in the exponent in the right-hand side of (2.2) by Proposition 3.3 of [5] and (2.3). We further note that condition (2.2) implies the inclusion {λ ∈ σ(A) Re λ ≥ −δ} ⊂ {λ ∈ C− | Im λ| ≥ c(− Re λ)−1/α } for some c, δ 0, see Proposition 3.7 of [5]. The next result from [7] provides the important converse of the above propo- sition for bounded semigroups on a Hilbert space, see Theorem 2.4 of [7]. Theorem 2.3. Let T (·) be a bounded C0-semigroup on a Hilbert space X such that σ(A) ⊂ C− and (2.2) holds for all λ ∈ iR. Then T (·) is polynomially stable of order α 0. For general Banach spaces X, in Theorem 5 in [6] this result was shown up to a logarithmic factor in the estimate in semigroup, see also [5], [15] and [17]. The paper [7] gives an example where this logarithmic correction actually occurs. Without assuming its boundedness, the semigroup is still polynomially stable if a holomorphic extension of R(λ, A)(ω − A)−α satisfies (2.3), but here one only obtains the stability order 2α + 1 + for any 0, see Proposition 3.4 of [5]. The proof of Theorem 2.3 is based on the following characterization of the boundedness of C0-semigroups on Hilbert spaces, see Theorem 2 in [10] and also Lemma 2.1 in [7]. Proposition 2.4. Let A generate the C0-semigroup T (·) on the Hilbert space X. The semigroup is bounded if and only if C+ ⊂ ρ(A) and sup r0 r R R(r + iτ, A)x2 + R(r + iτ, A∗ )x2 dτ ≤ c x2 for each x ∈ X. We now turn our attention to the concept of well-posed systems. From now on, X, U and Y are always Hilbert spaces, A generates the C0-semigroup T (·) on X and ω ω0(A). Let X−1 be the completion of X with respect to the norm given by x−1 = R(ω, A)x. We sometimes write XA −1 instead of X−1 to stress that this extrapolation space depends on A. The operator A has a unique extension A−1 ∈ B(X, X−1) which generates a C0-semigroup given by the continuous extension T−1(t) ∈ B(X−1) of T (t), t ≥ 0. We often omit the subscript −1 here. One can define such a space for each linear operator with non-empty resolvent set. Recall that we have set X1 = D(A). A bounded linear (observation) operator B : U −→ X−1 is called admissible for A (or the system (A, B, −) is called admissible) if the integral Φtu := t 0 T (t − s)Bu(s) ds
- 20. 6 E.M. Ait Benhassi, S. Boulite, L. Maniar and R. Schnaubelt belongs to X for all u ∈ L2 (0, t; U) and some t 0. (The integral is initially defined in X−1.) By Proposition 4.2.2 in [24], this property then holds for all t ≥ 0 and Φt ∈ B(L2 (0, t; U), X). Moreover, these operators are exponentially bounded, see Proposition 4.4.5 in [24]. A bounded linear (control) operator C : X1 −→ Y is called admissible for A (or the system (A, −, C) is called admissible) if the map Ψtx := CT (·)x, x ∈ X1, has a bounded extension in B(X, L2 (0, t; Y )) for some t 0. Propositions 4.2.3 and 4.3.3 in [24] show that this fact then holds for all t 0 and that the extensions are exponentially bounded. We still denote the extension by Ψt. One can extend an admissible observation operator C to the map CΛ given by CΛx = lim λ→∞ CλR(λ, A)x with domain D(CΛ) = {x ∈ X this limit exists in Y }. For each x ∈ X we have T (s)x ∈ D(CΛ) for a.e. s ≥ 0 and Ψtx = CΛT (·)x a.e. on [0, t] for all t 0 by, e.g., (5.6) and Proposition 5.3 in [28]. Theorem 4.4.3 of [24] shows that an operator B ∈ B(U, X−1) is admissible for A if and only if its adjoint B∗ ∈ B(D(A∗ ), U) is admissible for A∗ . Here we recall that X−1 is the dual space of D(A∗ ), if considered as a Banach space, see, e.g., Proposition 2.10.2 in [24]. Let system (A, B, C) be a system with a generator A and admissible control and observation operators B and C. One says that (A, B, C) is well posed if there are bounded linear operators Ft : L2 (0, t; U) −→ L2 (0, t; Y ) such that Fτ+tu = Fτ u1 on [0, τ], Ftu2 + ΨtΦτ u1 on [τ, τ + t] for all t, τ ≥ 0 and u ∈ L2 (0, τ + t; U), where u = u1 on (0, τ) and u = u2 on (τ, τ + t), see [27]. Also these (input-output) operators are exponentially bounded by Proposition 4.1 of [27]. One can introduce versions of the maps Ψt and Ft on the time interval R+ using L2 loc spaces. We denote these extensions by Ψ and F respectively. For x0 ∈ X and u ∈ L2 loc(R+, U) the output of the well-posed system (A, B, C) is then given by y = Ψx0 + Fu. In [27] it was shown that the Laplace transform ŷ of y satisfies ŷ(λ) = C(λ − A)−1 x0 + G(λ)û(λ) for all λ ∈ Cω, where G : Cω → B(U, Y ) is a bounded analytic function. It satisfies G (λ) = −CR(λ, A)2 B and it is thus determined by A, B and C up to an additive constant. (See, e.g., Theorem 2.7 in [22].) We call G the transfer function of (A, B, C). Set Z = D(A)+R(ω, A−1)BU and endow it with the norm zZ given by the infimum of all x1 + R(ω, A−1)Bv with z = x + R(ω, A−1)Bv, x ∈ D(A) and
- 21. Polynomial Internal and External Stability 7 v ∈ U. Theorem 3.4 and Corollary 3.5 of [22] then yield an extension C ∈ L(Z, U) of C such that the transfer function is represented as G(λ) = CR(λ, A−1)B + D, λ ∈ Cω, (2.4) for a feedthrough operator D ∈ L(U, Y ). Hence, the operators CR(λ, A−1)B are uniformly bounded on Cω. This representation of G is not unique in general since D(A) need not to be dense in Z. Under the additional assumption of regularity, one can replace here C by CΛ (possibly for a different D), see Theorem 5.8 in [27] and also Theorem 4.6 in [22] for refinements. We will not use regularity below. 3. Polynomial stabilizability and detectability In this section we introduce our new concepts and establish their basic properties. We start with the main definitions. Definition 3.1. The admissible system (A, B, −) is polynomially stabilizable (of or- der α 0) if there exists a generator ABF of a polynomially stable C0-semigroup TBF (·) on X (of order α 0) and an admissible observation operator F ∈ L(D(ABF ), U) of ABF such that R(λ, ABF ) = R(λ, A) + R(λ, A)BFR(λ, ABF ) (3.1) for all Re λ max{ω0(A), ω0(ABF )}. Definition 3.2. The admissible system (A, −, C) is polynomially detectable (of order α 0) if there exists a generator AHC of a polynomially stable C0-semigroup THC(·) (of order α 0) and an admissible control operator H ∈ L(Y, XAHC −1 ) of AHC such that R(λ, AHC ) = R(λ, A) + R(λ, (AHC )−1)HCR(λ, A) (3.2) for all Re λ max{ω0(A), ω0(AHC )}. Here F, resp. H, plays the role of a feedback. These definitions are inspired by the Definition 3.2 in [12] for the exponentially stable case. For this case, in, e.g., [29] concepts of exponential stabilizability or detectability were used which are (at least formally) a bit stronger than those in [12], cf. Remark 3.3(b). In our context, one could also include the boundedness of the feedback semigroup TBF (·) or THC (·) in the above definitions since the theory of polynomial stability works much better in the bounded case, as seen in the previous section. Instead, we make additional boundedness assumptions in some of our results. In applications one can check the boundedness or TBF (·) or THC (·) by showing that the generators ABF or AHC are dissipative, respectively, where one may use their representation given in the next remark.
- 22. 8 E.M. Ait Benhassi, S. Boulite, L. Maniar and R. Schnaubelt Remark 3.3. (a) Let (A, B, −), (ABF , −, F), (A, −, C) and (AHC , H, −) be admis- sible. Proposition 4.11 in [13] (with β = γ = 1 and b = c = 0) then shows that the equations (3.1) and (3.2) are equivalent to TBF (t)x = T (t)x + t 0 T (t − s)BFΛTBF (s)x ds = T (t)x + ΦtFΛTBF (·)x, (3.3) THC(t)x = T (t)x + t 0 THC(t − s)HCΛT (s)x ds (3.4) for all t ≥ 0 and x ∈ X, respectively. (b) Applying λ − A−1 to (3.1), we see that ABF is a restriction of the part (A−1 +BF)|X of A−1 +BF in X. Similarly, multiplication of (3.2) by λ−AHC,−1 leads to A ⊂ (AHC,−1 − HC)|X. See Proposition 6.6 in [28]. We note that in [29] exponential stabilizability and detectability was defined in such a way that ABF = (A−1 + BFΛ)|X and AHC = (A−1 + CHΛ)|X. (c) The system (A, B, −) is polynomially stabilizable of order α 0 (with feedback F) if and only if (A∗ , −, B∗ ) is polynomially detectable of order α 0 (with feedback H = F∗ ). Moreover, the semigroups of the feedback systems are dual to each other. (d) Let L be a closed operator with ∅ = Λ ⊂ ρ(L) and Ω ⊃ Λ be connected. If R(·, L) has a holomorphic extension Rλ to Ω, then Ω ⊂ ρ(L) and Rλ = R(λ, L) for every λ ∈ Ω. (See Proposition B5 in [3].) In a sequence of lemmas we relate the growth properties of several operators arising in (3.1) or (3.2). We use the spectral bound s(L) = sup{Re λ λ ∈ σ(L)} ∈ [−∞, ∞] for a closed operator L, where sup ∅ = −∞ Lemma 3.4. Let C ∈ B(X1, Y ) and B ∈ B(U, X−1) be admissible observation and control operators for A, respectively and let R(r + iτ, A) ≤ c |τ|α (3.5) for some r s(A) and α 0 and all |τ| ≥ 1. We then obtain the estimates CR(r + iτ, A) ≤ c |τ|α and R(r + iτ, A)B ≤ c |τ|α for all |τ| ≥ 1. Moreover, if (A, B, C) is also well posed, we have CR(r + iτ, A)B ≤ c |τ|α for all |τ| ≥ 1. Here the constants are uniform for r in bounded intervals. Proof. Let λ = r + iτ and μ = ω + iτ for τ ∈ R and some ω max{0, ω0(A)}. The resolvent equation yields CR(λ, A) = CR(μ, A) + (ω − r)CR(μ, A)R(λ, A). (3.6) Let x ∈ D(A). Since the resolvent is the Laplace transform of T (·), from the admissibility of C and exponential bound of T (·) we deduce CR(μ, A)x2 ≤ ∞ 0 e− ω 2 t e− ω 2 t CT (t)x dt 2 ≤ c ∞ 0 e−ωt CT (t)x2 dt (3.7)
- 23. Polynomial Internal and External Stability 9 ≤ c ∞ n=0 e−ωn CT (·)T (n)x2 L2(0,1;Y ) ≤ c ∞ n=0 e−ωn T (n)x2 ≤ c x2 . By density, the formulas (3.5), (3.6) and (3.7) imply CR(λ, A) ≤ c + c |τ|α ≤ c |τ|α for |τ| ≥ 1. The second asserted inequality then follows by duality because B∗ is an admissible observation operator for A∗ and R(λ, A)B = B∗ R(λ, A∗ ). For the final claim, we start from the equation CR(λ, A)B = CR(μ, A)B + (ω − r)CR(μ, A)R(λ, A)B for λ = r + iτ, μ = ω + iτ, τ ∈ R and some ω max{0, ω0(A)}. As noted in the previous section, CR(μ, A)B : U → Y is uniformly bounded. The third assertion now is a consequence of the two previous ones. In the next lemma we deduce resolvent estimates for A from those for ABF . Lemma 3.5. Let B ∈ L(U, X−1) be an admissible control operator for A. Assume that there exist a generator ABF of a C0-semigroup TBF (·) on X and an admissible observation operator F ∈ L(D(ABF ), U) of ABF such that (3.1) holds. Assume that R(λ, ABF ) ≤ c (1 + |λ|α ) for r Re λ ≤ r + δ and some r ≥ s(ABF ), δ 0, α ≥ 0. Suppose that R(λ, A)B has a holomorphic extension RB λ to Cr satisfying RB λ ≤ c (1 + |λ|β ) for r Re λ ≤ r + δ and some β ≥ 0. Then R(·, A) can be extended to a neigh- borhood of Cr, and we obtain R(λ, A) ≤ c (1 + |λ|α+β ) (3.8) for r ≤ Re λ ≤ r + δ. Moreover, (3.1) holds on Cr. If r = 0, then T (·) is polyno- mially stable with order 2(α + β) + 1 + η for any η 0. Proof. By the assumption, (3.1) and Remark 3.3, the resolvent R(·, A) has the extension R(λ, A) = R(λ, ABF ) − RB λ FR(λ, ABF ) to λ ∈ Cr. Lemma 3.4 and the assumption then imply that R(λ, A) ≤ c (1 + |λ|α+β ) for r Re λ ≤ r + δ. A standard power series argument allows us to extend this inequality to λ ∈ Cr and to deduce that a neighborhood of Cr belongs to ρ(A). The uniqueness of the holomorphic extension now yields that RB λ = R(λ, A)B on Cr and that (3.1) holds on Cr. The last assertion then follows from estimate (3.8) and Propositions 3.4 and 3.6 in [5]. The next result is proved in the same manner as the above lemma.
- 24. 10 E.M. Ait Benhassi, S. Boulite, L. Maniar and R. Schnaubelt Lemma 3.6. Let the operators A, C and H satisfy the assumptions of Definition 3.2 except for the polynomial stability of THC(·). Assume that R(λ, AHC) ≤ c (1 + |λ|α ) for r Re λ ≤ r + δ and some r ≥ s(AHC ), δ 0 and α ≥ 0. Let CR(λ, A) have a holomorphic extension RC λ to Cr. Suppose that RC λ ≤ c (1 + |λ|β ) for r Re λ ≤ r + δ and some β 0. Then ρ(A) contains a neighborhood of Cr, the equality (3.2) holds on Cr, and we obtain R(λ, A) ≤ c (1 + |λ|α+β ) for r ≤ Re λ ≤ r + δ. If r = 0, then T (·) is polynomially stable with order 2(α + β) + 1 + η for any η 0. To apply Proposition 2.4, we will need a variant of the above estimates. Lemma 3.7. Let A generate a bounded C0-semigroup and C be an admissible ob- servation operator for A. Then sup r0 r R CR(r + iτ, A)x2 dτ ≤ c x2 for all r 0 and x ∈ X. Proof. Take r 0 and x ∈ D(A). Since A − r generates the exponentially stable semigroup (e−rt T (t))t≥0, Plancherel’s theorem and the assumption yield CR(r + i·, A)x2 L2(R+,Y ) = Ce−r· T (·)x2 L2(R+,Y ) = n≥0 1 0 e−2rn e−2rs CT (s)T (n)x2 ds. ≤ c n≥0 e−2rn T (n)x2 ≤ c x2 1 − e−2r ≤ c r x2 . The assertion follows by density. 4. Main results We show that external polynomial stability in the frequency domain, i.e., a poly- nomial estimate on the transfer function, imply polynomial stability of the state system. We begin with a result involving only the control operator B. Proposition 4.1. Let (A, B, −) be admissible and polynomially stabilizable of order α 0. Assume that R(λ, A)B has a holomorphic extension to C+ which is bounded by c (1 + |λ|β ) for 0 Re λ ≤ δ and some β ≥ 0, δ 0. The following assertions hold.
- 25. Polynomial Internal and External Stability 11 a) The resolvent R(·, A) can be extended to a neighborhood of C+ and R(λ, A) ≤ cε (1 + |λ|α+β+ε ) (4.1) for 0 ≤ Re λ ≤ δ and every ε 0. If TBF (·) is bounded, we can choose ε = 0. b) The semigroup T (·) is polynomially stable. If T (·) is also bounded, then it is polynomially stable of order α + β + ε. If in addition TBF (·) is bounded, we can take ε = 0. Proof. a) Propositions 3.3 and 3.6 in [5] imply that σ(ABF ) ⊂ C− and R(λ, ABF ) ≤ cε(1 + |λ|α+ε ) holds for Re λ ≥ 0 and every ε 0. Using Lemma 3.5, we infer σ(A) ⊂ C− and (4.1). If TBF (·) is bounded, we can use Proposition 2.2 instead of the results from [5] and obtain the above estimates with ε = 0. b) Proposition 3.4 of [5] and (4.1) imply the polynomial stability of T (·). If also T (·) is bounded, it is polynomially stable of order α+β+ε due to Theorem 2.3 and (4.1). By duality, the above proposition implies the next one for the observation system (A, −, C). Proposition 4.2. Let (A, −, C) be admissible and polynomially detectable of order α 0. Assume that CR(·, A) has a holomorphic extension to C+ which is bounded by c (1 + |λ|β ) for 0 Re λ ≤ δ and some β ≥ 0. The following assertions hold. a) The resolvent R(·, A) can be extended to a neighborhood of C+ and estimate (4.1) holds for every ε 0. If THC(·) is bounded, we can take ε = 0. b) The semigroup T (·) is polynomially stable. If T (·) is also bounded, then it is polynomially stable of order α + β + ε. If in addition THC (·) is bounded, we can take ε = 0. We now can state our main result which uses the full system (A, B, C) and the transfer function G. Theorem 4.3. Let (A, B, C) be a well-posed system which is polynomially stabiliz- able of order α 0 and polynomially detectable of order β 0. Assume that G has a holomorphic extension to C+ which is bounded by c (1 + |λ|γ ) for 0 Re λ ≤ δ and some γ ≥ 0 and δ 0. The following assertions hold. a) The extension C of C is an admissible observation operator for ABF , σ(A) ⊂ C−, and R(λ, A) ≤ cε(1 + |λ|α+β+γ+ε ) for 0 Re λ ≤ δ and all ε 0. If TBF (·) is bounded, we can take ε = 0. b) The semigroup T (·) is polynomially stable. If T (·) is bounded, then it is poly- nomially stable of order α + β + γ + ε. If in addition TBF (·) is bounded, we can take ε = 0.
- 26. 12 E.M. Ait Benhassi, S. Boulite, L. Maniar and R. Schnaubelt Proof. a) Due to (3.1) and (2.4), we have D(ABF ) ⊂ Z and CR(λ, ABF ) = CR(λ, A) + CR(λ, A)BFR(λ, ABF ), CR(λ, ABF ) = CR(λ, A) + G(λ)FR(λ, ABF ) − DFR(λ, ABF ) (4.2) for Re λ max{ω0(A), ω0(ABF )}. Taking the inverse Laplace transform of this equation, we define ΨBF x := L−1 (CR(·, ABF )x) = Ψx + FFTBF (·)x − DFTBF (·)x (4.3) for x ∈ D(ABF ). By assumption, ΨBF : X −→ L2 loc(R+, Y ) is continuous. For τ ≥ 0 and x ∈ D(ABF ), the properties of a well-posed system and (3.3) yield ΨBF x(· + τ) = ΨT (τ)x + FFTBF (·)TBF (τ)x + ΨΦτ FTBF (·)x − DFTBF (·)TBF (τ)x = ΨTBF (τ)x + FFTBF (·)TBF (τ)x − DFTBF (·)TBF (τ)x = ΨBF TBF (τ)x. As a result, (ΨBF , TBF ) is an observation system in the sense of [26] or Sec- tion 4.3 in [24]. The proof of Theorem 3.3 of [26] and (4.3) thus show that ΨBF x = CTBF (·)x for x ∈ D(ABF ) and the admissible control operator C ∈ L(D(ABF ), Y ) for ABF given by Cx = ΨBF (λ)(λ − ABF )x = CR(λ, ABF )(λ − ABF )x = Cx for x ∈ D(ABF ); i.e., ΨBF x = CTBF (·)x for x ∈ D(ABF ). Proposition 3.4 of [5] and Lemma 3.4 then yield CR(λ, ABF ) ≤ c (1 + |λ|α+ε ) and FR(λ, ABF ) ≤ c (1 + |λ|α+ε ) for Re λ ≥ 0 and any ε 0. If TBF (·) is bounded, we can use Proposition 2.2 instead of the results in [5] and derive these estimates with ε = 0. By means of (4.2) and the bound on G, we now extend CR(·, A) (using the same symbol) to C+ and obtain CR(λ, A) ≤ c (1 + |λ|α+γ+ε ) for 0 Re λ ≤ δ. Proposition 4.2 then gives R(λ, A) ≤ cε(1 + |λ|α+β+γ+ε ) for 0 Re λ ≤ δ and all ε 0, where we can take ε = 0 if TBF (·) is bounded. b) Proposition 3.4 of [5] and part a) imply the polynomial stability of T (·). If T (·) is bounded, it is polynomially stable of order α+β +γ +ε due to Theorem 2.3 and part a), where we can take ε = 0 if TBF (·) is bounded. In the above results one obtains the expected stability order of T (·) only if this semigroup is bounded. This property automatically holds in the important case of a scattering passive system (A, B, C); i.e., if we have y2 L2(0,t;Y ) + x(t)2 ≤ u2 L2(0,t;U) + x02
- 27. Polynomial Internal and External Stability 13 for all u ∈ L2 (0, t; U), x0 ∈ X and t ≥ 0, where x(t) = T (t)x0 + Φtu is the state and y = Ψx0 + Fu is the output of (A, B, C). This class of systems has been characterized and studied in, e.g., [22]. In this case T (t) and G(λ) are contractions for t ≥ 0 and λ ∈ C+ by Proposition 7.2 and Theorem 7.4 of [22]. Corollary 4.4. Let (A, B, C) be a scattering passive system which is polynomially stabilizable of order α 0 and polynomially detectable of order β 0. Then T (·) is polynomially stable of order α + β + ε for each ε 0. We can take ε = 0 if TBF (·) is bounded. Proposition 2.4 yields another sufficient condition for the boundedness of T (·) in the framework of the first two propositions of this section. Proposition 4.5. Assume that the assumptions of both Propositions 4.1 and 4.2 hold for some α 0 and for β = 0. Let TBF (·) and THC(·) be bounded. Then T (·) is bounded, and hence polynomially stable of order α 0. Proof. Definitions 3.1 and 3.2 yield R(r + iτ, A)x = R(r + iτ, ABF )x − R(r + iτ, A)BFR(r + iτ, ABF )x, (4.4) R(r+iτ, A∗ )x = R(r + iτ, A∗ HC )x − R(r + iτ, A∗ )C∗ H∗ R(r+iτ, A∗ HC)x (4.5) for all r max{ω0(A), 0}, τ ∈ R and x ∈ X. We can extend these equations to r 0 using the bounded extensions of R(λ, A)B and R(λ, A∗ )C∗ = (CR(λ, A))∗ which are provided by our assumption. Since TBF (·) and THC(·) are bounded, Lemma 3.7 implies that the terms on the right-hand sides belong to L2 (R, X) as functions in τ, with norms bounded by cr−1/2 x. Employing Proposition 2.4, we then deduce the boundedness of T (·) from (4.4) and (4.5). The final assertion now follows from Proposition 4.1. We finally present sufficient conditions for polynomial stabilizability and for polynomial detectability by means of a decomposition into a polynomial stable and an observable part. An admissible system (A, B, −) is called null controllable in finite time if for each initial value x0 ∈ X there is a time τ 0 and a control u ∈ L2 (0, τ; U) such that x(τ) = T (τ)x0 + Φτ u = 0. We further note that one can extend an operator S to X−1 if it commutes with T (t) for all t ≥ 0 since then SR(ω, A) = R(ω, A)S. Theorem 4.6. Let (A, B, −) be admissible and let P2 = P ∈ B(X) satisfy T (t)P = PT (t) for all t ≥ 0. Set Xs = PX, Xu = (I −P)X, Ts(t) = T (t)P, Au = (I −P)A and Bu = (I − P)B. Assume that (i) the C0-semigroup Ts(·) is polynomially stable of order α 0 on Xs and (ii) the system (Au, Bu, −) is null controllable in finite time on Xu. Then the system (A, B, −) is polynomially stabililizable of order α 0. Proof. First observe that Tu(·) is the C0-semigroup on Xu generated by Au and that Bu is admissible for Au. Due to (ii), for each x0 ∈ Xu there is a time τ 0 and a control u ∈ L2 (0, τ; U) such that xu(τ) = Tu(τ)x0 + (I − P)Φτ u = 0. Extending
- 28. 14 E.M. Ait Benhassi, S. Boulite, L. Maniar and R. Schnaubelt xu and u by 0 to (τ, ∞), we see that the system (Au, Bu, −) is optimizable in the sense of Definition 3.1 in [29]. Propositions 3.3 and 3.4 of [29] (or Theorem 2.2 of [9]) then give an operator Fu which satisfies the conditions of Definition 3.1 where TBuFu (·) is even exponentially stable, i.e., ω0(ABuFu ) 0. We thus have R(λ, ABuFu ) = R(λ, Au) + R(λ, Au)BuFuR(λ, ABuFu ) (4.6) for all Re λ max(ω0(A), ω0(ABuFu )). We now set F = 0 Fu and ABF := As 0 0 ABuFu . It is then straightforward to check that these operators fulfill the conditions of Definition 3.1. The next result follows by duality from Theorem 4.6. Theorem 4.7. Let (A, −, C) be admissible and let P2 = P ∈ B(X) satisfy T (t)P = PT (t) for all t ≥ 0. Set Xs = PX, Xu = (I −P)X, Ts(t) = T (t)P, Au = (I −P)A and Cu = C(I − P). Assume that (i) the C0-semigroup Ts(·) is polynomially stable of order α 0 on Xs and (ii) the system (A∗ u, C∗ u, −) is null controllable in finite time on Xu. Then the system (A, −, C) is polynomially detectable of order α 0. Remark 4.8. The results of Theorem 4.6 and 4.7 also hold if we replace the con- dition (ii) by (ii) : The system (Au, Bu, −) (resp., (A∗ u, C∗ u, −)) is polynomially stabilizable of order α. References [1] F. Alabau, P. Cannarsa and V. Komornik, Indirect internal stabilization of weakly coupled evolution equations. J. Evol. Equ. 2 (2002), 127–150. [2] K. Ammari and M. Tucsnak, Stabilization of second-order evolution equations by a class of unbounded feedbacks. ESAIM Control Optim. Calc. Var. 6 (2001), 361–386. [3] W. Arendt, C.J.K. Batty, M. Hieber and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems. Birkhäuser Verlag, Basel, 2001. [4] G. Avalos and R. Triggiani, Rational decay rates for a PDE heat-structure interac- tion: a frequency domain approach. Evol. Equ. Control Theory 2 (2013), 233–253. [5] A. Bátkai, K.-J. Engel, J. Prüss and R. Schnaubelt, Polynomial stability of operator semigroups. Math. Nachr. 279 (2006), 1425–1440. [6] C.J.K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces. J. Evol. Eq. 8 (2008), 765–780. [7] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups. Math. Ann. 347 (2010), 455–478. [8] N. Burq and M. Hitrik, Energy decay for damped wave equations on partially rect- angular domains. Math. Res. Lett. 14 (2007), 35–47.
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- 30. 16 E.M. Ait Benhassi, S. Boulite, L. Maniar and R. Schnaubelt [27] G. Weiss, Transfer functions of regular linear systems. Part I: Characterization of regularity, Trans. Amer. Math. Soc. 342 (1994), 827–854. [28] G. Weiss, Regular linear systems with feedback. Math. Control Signals Systems 7 (1994), 23–57. [29] G. Weiss and R. Rebarber, Optimizability and estimatability for infinite-diemnsional systems. SIAM J. Control Optim. 39 (2000), 1204–1232. El Mustapha Ait Benhassi and Lahcen Maniar Cadi Ayyad University Faculty of Sciences Semlalia B.P. 2390 Marrakesh, Morocco e-mail: m.benhassi@ucam.ac.ma maniar@ucam.ac.ma Said Boulite Hassan II University Faculty of Sciences Ain Chock B.P. 5366 Maarif 20100 Casablanca, Morocco e-mail: s.boulite@fsac.ac.ma Roland Schnaubelt Department of Mathematics Karlsruhe Institute of Technology D-76128 Karlsruhe, Germany e-mail: schnaubelt@kit.edu
- 31. Operator Theory: Advances and Applications, Vol. 250, 17–29 Minimal Primal Ideals in the Multiplier Algebra of a C0(X)-algebra R.J. Archbold and D.W.B. Somerset Abstract. Let A be a stable, σ-unital, continuous C0(X)-algebra with sur- jective base map φ : Prim(A) → X, where Prim(A) is the primitive ideal space of the C∗ -algebra A. Suppose that φ−1 (x) is contained in a limit set in Prim(A) for each x ∈ X (so that A is quasi-standard). Let CR(X) be the ring of continuous real-valued functions on X. It is shown that there is a homeo- morphism between the space of minimal prime ideals of CR(X) and the space MinPrimal(M(A)) of minimal closed primal ideals of the multiplier algebra M(A). If A is separable then MinPrimal(M(A)) is compact and extremally disconnected but if X = βN N then MinPrimal(M(A)) is nowhere locally compact. Mathematics Subject Classification (2010). Primary 46L05, 46L08, 46L45; Sec- ondary 46E25, 46J10, 54C35. Keywords. C∗ -algebra, C0(X)-algebra, multiplier algebra, minimal prime ideal, minimal primal ideal, primitive ideal space, quasi-standard. 1. Introduction Let A be a C∗ -algebra with multiplier algebra M(A) [10] and with primitive ideal space Prim(A). The ideal structure of M(A) has been widely studied, and is typi- cally much more complicated than that of A, see for example [1], [13], [21], [25],[27]. One approach, which the authors used in an earlier paper [7], is to endow A with a C0(X)-structure (this can always be done, sometimes in many different ways). Let A be a σ-unital C0(X)-algebra (defined below) with base map φ : Prim(A) → X, and let Xφ denote the image of Prim(A) under φ. The authors showed that there is a map from the lattice of z-ideals of CR(Xφ) into the lattice of closed ideals of M(A), and that this map is injective if A is stable [7, Theorem 3.2]. If Xφ is infinite then z-ideals generally exist in great profusion – for example, CR(R) has uncountable chains of prime z-ideals associated with each point of R [22], [26] – c Springer International Publishing Switzerland 2015
- 32. 18 R.J. Archbold and D.W.B. Somerset so this yields a vast multiplicity of closed ideals in M(A) and indicates something of the complexity of Prim(M(A)) [7, Theorem 5.3]. The most studied z-ideals are the minimal prime ideals and in this note we consider the image of the space of minimal prime ideals of CR(Xφ) under the injective map. We show in Theorem 3.4 that if A is stable, σ-unital, and quasi-standard (defined below) then the image of the space of minimal prime ideals is precisely MinPrimal(M(A)), the space of minimal closed primal ideals of M(A) (see below). It follows that MinPrimal(M(A)) is totally disconnected and countably compact (Corollary 4.1). If A is also separable – for example if A equals C[0, 1] ⊗ K(H) (where K(H) is the algebra of compact operators on a separable infinite-dimensional Hilbert space) – then MinPrimal(M(A)) is compact and extremally disconnected (Corollary 4.3). All ideals in this paper will be two-sided, but not necessarily closed unless stated to be so. An ideal J in a C∗ -algebra A is primal if whenever I1, . . . , In is a finite collection of ideals of A with the product I1 . . . In = {0} then Ii ⊆ J for at least one i ∈ {1, . . . , n}. An equivalent definition, when J is closed, is that the hull of J should be contained in a limit set in Prim(A) [3, Proposition 3.2]. Every primitive ideal is prime and hence primal. Each closed primal ideal of a C∗ -algebra A contains one or more minimal closed primal ideals [2, p. 525]. The space of minimal closed primal ideals with the τw-topology (defined in Section 3) is denoted MinPrimal(A). This Hausdorff space is often identifiable in situations where the primitive ideal space is non-Hausdorff and highly complicated. Indeed, the multiplier algebras considered in this paper are a case in point. A C∗ -algebra A is said to be quasi-standard if the relation ∼ of inseparability by disjoint open sets is an open equivalence relation on Prim(A) [5]. This condition is a wide generalisation of the special case where Prim(A) is Hausdorff. Examples include, in the unital case, von Neumann and AW∗ -algebras, local multiplier alge- bras of C∗ -algebras [29], and the group C∗ -algebras of amenable discrete groups [17]; and in the non-unital case, many other group C∗ -algebras, see [4]. A basic non-unital example, however, is simply A = C0(X) ⊗ K(H), where X is a locally compact Hausdorff space, and even in this case the ideal structure of M(A) is not well understood, see [20], [7]. The connection between quasi-standard C∗ -algebras and C0(X)-algebras is explained in Lemma 2.1 and the remarks preceding it. The structure of the paper is that in Section 2, we set up some machinery; in Section 3, we prove the main homeomorphism result; and in Section 4, we give some applications. 2. Preliminaries First we collect the information that we need on C0(X)-algebras. Recall that a C∗ -algebra A is a C0(X)-algebra if there is a continuous map φ, called the base map, from Prim(A), the primitive ideal space of A with the hull-kernel topology, to the locally compact Hausdorff space X [31, Proposition C.5]. Then Xφ, the
- 33. Minimal Primal Ideals in the Multiplier Algebra 19 image of φ in X, is completely regular; and if A is σ-unital, Xφ is σ-compact and hence normal [7, Section 1]. If φ is an open map then Xφ is locally compact. For x ∈ Xφ, set Jx = {P ∈ Prim(A) : φ(P) = x}, and for x ∈ X Xφ, set Jx = A. For a ∈ A, the function x → a + Jx (x ∈ X) is upper semi-continuous [31, Proposition C.10]. The C0(X)-algebra A is said to be continuous if, for all a ∈ A, the norm function x → a + Jx (x ∈ X) is continuous. By Lee’s theorem [31, Proposition C.10 and Theorem C.26], this happens if and only if the base map φ is open. An important special case (through which all other cases factor) is when φ is the complete regularization map φA for Prim(A) [14, Theorem 3.9]. In this case, the ideals Jx (x ∈ Xφ) are called the Glimm ideals of A, and the set of Glimm ideals with the complete regularization topology is called Glimm(A). Each minimal closed primal ideal of A contains a unique Glimm ideal [5, Lemma 2.2]. If A is quasi-standard then the complete regularization map φA is open [5, Theorem 3.3], so Glimm(A) is locally compact and A is a continuous C0(X)-algebra with X = XφA = Glimm(A). Furthermore, if A is quasi-standard then each Glimm ideal of A is actually primal and indeed the topological spaces Glimm(A) and MinPrimal(A) coincide [5, Theorem 3.3]. It then follows from [3, Proposition 3.2] that φ−1 A (x) is a maximal limit set in Prim(A) for all x ∈ X. The following result is closely related to [5, Theorem 3.4]. Lemma 2.1. For a C∗ -algebra A, the following are equivalent: (i) A is quasi-standard; (ii) A is a continuous C0(X)-algebra over a locally compact Hausdorff space X with base map φ such that φ−1 (x) is contained in a limit set in Prim(A) for all x ∈ Xφ. When these equivalent conditions hold, there is a homeomorphism ψ : Glimm(A) → Xφ such that φ = ψ◦φA, where φA is the complete regularization map for A. More- over, for all x ∈ Xφ, φ−1 (x) is a maximal limit set in Prim(A) and Jx is a minimal closed primal ideal of A. Proof. We have seen that (i) implies (ii). Conversely, suppose that (ii) holds. Since X is a locally compact Hausdorff space, for P, Q ∈ Prim(A), P ∼ Q if and only if φ(P) = φ(Q). It follows that ∼ is an equivalence relation. Let Y be a non-empty open subset of Prim(A). Then Y := φ−1 (φ(Y )) is the ∼-saturation of Y , and Y is open since φ is open. Hence ∼ is an open equivalence relation so (i) holds. When (ii) holds, we have that φ is continuous and open with image Xφ, and that it factors as φ = ψ ◦ φA, where ψ : Glimm(A) → Xφ is continuous [14, Theorem 3.9]. Then ψ is surjective, and the limit set hypothesis easily shows that ψ is injective. Since φ is open and φA is continuous, ψ is open. Thus ψ is a homeomorphism. Finally, let x ∈ Xφ and let Ω be a net in Prim(A) whose limit set L contains φ−1 (x). Since φA is constant on L, φ(L) = {x}. Thus L = φ−1 (x) and φ−1 (x) is a maximal limit set. It follows from [3, Proposition 3.2] that Jx is a minimal closed primal ideal of A.
- 34. 20 R.J. Archbold and D.W.B. Somerset Now let J be a proper, closed ideal of a C∗ -algebra A. The quotient map qJ : A → A/J has a canonical extension ˜ qJ : M(A) → M(A/J) such that ˜ qJ (b)qJ (a) = qJ (ba) and qJ (a) ˜ qJ (b) = qJ (ab) (a ∈ A, b ∈ M(A)). We define a proper, closed ideal ˜ J of M(A) by ˜ J = ker ˜ qJ = {b ∈ M(A) : ba, ab ∈ J for all a ∈ A}. Various properties of ˜ J were established in [6, Proposition 1.1]. For example, ˜ J is the strict closure of J in M(A) and ˜ J ∩ A = J. The following proposition was proved in [6, Proposition 1.2]. Proposition 2.2. Let A be a C0(X)-algebra with base map φ. Then φ has a unique extension to a continuous map φ : Prim(M(A)) → βX such that φ(P̃) = φ(P) for all P ∈ Prim(A). Hence M(A) is a C(βX)-algebra with base map φ and Im(φ) = clβX(Xφ). Now let A be a C0(X)-algebra with base map φ and let φ : Prim(M(A)) → βX be as in Proposition 2.2. For x ∈ βX, we define Hx = {Q ∈ Prim(M(A)) : φ(Q) = x}, a closed two-sided ideal of M(A). Thus Hx is defined in relation to (M(A), βX, φ) in the same way that Jx (for x ∈ X) is defined in relation to (A, X, φ). It fol- lows that for each b ∈ M(A), the function x → b + Hx (x ∈ βX) is up- per semi-continuous. If φ is the complete regularization map for Prim(A) and X = βGlimm(A) then Glimm(M(A)) = {Hx : x ∈ X}; see the comment after [9, Proposition 4.4]. The next proposition is contained in [7, Proposition 2.3]. Proposition 2.3. Let A be a C0(X)-algebra with base map φ, and set Xφ = Im(φ). (i) For all x ∈ X, Jx ⊆ Hx ⊆ ˜ Jx and Jx = Hx ∩ A. (ii) For all b ∈ M(A), b = sup{b + ˜ Jx : x ∈ Xφ} = sup{b + Hx : x ∈ Xφ}. In the case when A = C0(X)⊗K(H) ∼ = C0(X, K(H)) and φ : Prim(A) → X is the homeomorphism such that φ−1 (x) = {f ∈ C0(X) : f(x) = 0} ⊗ K(H) (x ∈ X), the multiplier algebra M(A) is isomorphic to the C∗ -algebra of bounded, strong∗ - continuous functions from X to B(H) (the algebra of bounded linear operators on the Hilbert space H) [1, Corollary 3.5]. Then for x ∈ X, ˜ Jx = {f ∈ M(A) : f(x) = 0}. On the other hand, by Proposition 2.2 M(A) is a C(βX)-algebra, and for x ∈ βX, Hx = {f ∈ M(A) : lim y→x f(y) = 0}.
- 35. Minimal Primal Ideals in the Multiplier Algebra 21 We shall recall in Theorem 2.4 below that when A is a σ-unital C0(X)-algebra with base map φ there is an order-preserving map from the lattice of z-ideals of CR(Xφ) into the lattice of closed ideals of M(A). To describe this map, we give a brief account of the theory of z-ideals. Let X be a completely regular topological space and let CR(X) denote the ring of continuous real-valued functions on X. For f ∈ CR(X), let Z(f) = {x ∈ X : f(x) = 0}, the zero set of f. We note for later that every zero set clearly arises as the zero set of a bounded continuous function. A non-empty family F of zero sets of X is called a z-filter if: (i) F is closed under finite intersections; (ii) ∅ / ∈ F; (iii) each zero set which contains a member of F belongs to F. Each ideal I ⊆ CR(X) yields a z-filter Z(I) = {Z(f) : f ∈ I}. An ideal I is called a z-ideal if Z(f) ∈ Z(I) implies f ∈ I; and if F is a z-filter on X then the ideal I(F) defined by I(F) = {f ∈ CR(X) : Z(f) ∈ F} is a z-ideal. There is a bijective correspondence between the set of z-ideals of CR(X) and the set of z-filters on X, given by I = I(Z(I)) ↔ Z(I). Now let A be a σ-unital C0(X)-algebra with base map φ, and let u ∈ A be a strictly positive element. For a ∈ A, set Z(a) = {x ∈ Xφ : a ∈ Jx}. Unless norm functions of elements of A are continuous on Xφ, Z(a) will not necessarily be a zero set of Xφ. However, since Z(u) = ∅ and A is closed under multiplication by Cb (Xφ), every zero set Z(f) of Xφ arises as Z(a) for the element a = f · u ∈ A (f ∈ Cb R(Xφ)). For b ∈ M(A), set Z(b) = {x ∈ Xφ : b ∈ ˜ Jx}. Note that if b ∈ A then this definition is consistent with the previous one because ˜ Jx ∩ A = Jx (x ∈ Xφ). It is also useful to note that for b ∈ M(A) and x ∈ Xφ, b ∈ ˜ Jx if and only if bu ∈ ˜ Jx if and only if bu ∈ Jx. Hence Z(b) = Z(bu). For a z-filter F on Xφ define Lalg F = {b ∈ M(A) : ∃Z ∈ F, Z(b) ⊇ Z}, and let LF be the norm-closure of Lalg F in M(A). Let b ∈ Lalg F . Then for a ∈ M(A), Z(ab) ⊇ Z(b) and Z(ba) ⊇ Z(b), while for a ∈ Lalg F , Z(a+ b) ⊇ Z(a)∩Z(b). Hence Lalg F is an ideal of M(A), so LF is a closed ideal of M(A). Theorem 2.4. ([7, Theorem 3.2]) Let A be a σ-unital C0(X)-algebra with base map φ. Suppose that A/Jx is non-unital for all x ∈ Xφ. Let I and J be z-ideals of CR(Xφ) and suppose that there exists a zero set Z of Xφ such that Z ∈ Z[I] but Z / ∈ Z[J]. Then LZ[I] ⊆ LZ[J]. Hence the assignment I → LZ[I] defines an order- preserving injective map L from the lattice of z-ideals of CR(Xφ) into the lattice of closed ideals of M(A). To identify what happens to some of the most important z-ideals of CR(Xφ) under this map, we use the following notation. For x ∈ X, let Mx be the maximal ideal given by Mx = {f ∈ CR(X) : f(x) = 0}, and let Ox = {f ∈ CR(X) : x ∈ int(Z(f))}
- 36. 22 R.J. Archbold and D.W.B. Somerset where int(Z(f)) denotes the interior of Z(f). Then Mx and Ox are z-ideals, and Ox is the smallest ideal of CR(X) which is not contained in any maximal ideal other than Mx. The definitions just given can be extended as follows. For p ∈ βX, let Mp = {f ∈ CR(X) : p ∈ clβXZ(f)} and define Op to be the set of all f ∈ CR(X) for which clβXZ(f) is a neighbourhood of p in βX. Then for x ∈ X, Mx = Mx and Ox = Ox. The embedding map takes Mx to ˜ Jx and Op to Hp (and hence Ox to Hx). Proposition 2.5 ([7, Theorem 4.3]). Let A be a σ-unital C0(X)-algebra with base map φ. (i) For x ∈ Xφ, LZ[Mx] = ˜ Jx. (ii) For p ∈ clβXXφ, LZ[Op] = Hp. Proposition 2.5 shows that the embedding map of Theorem 2.4 is mainly shedding light on the lattice of closed ideals of M(A) between ˜ Jx and Hx; see [7, Section 4] for further discussion. Before presenting a simple example to illustrate Theorem 2.4 and Proposition 2.5, we need further terminology. A z-filter F on a completely regular space X is said to be prime if Z1 ∪Z2 ∈ F implies that either Z1 ∈ F or Z2 ∈ F, for zero sets Z1 and Z2. Let PF(X) denote the set of prime z-filters, and let PZ(X) be the set of prime z-ideals (recall that an ideal P ⊆ CR(X) is prime if fg ∈ P implies f ∈ P or g ∈ P). The bijective correspondence between z-ideals and z-filters restricts to a bijective correspondence j : PZ(X) → PF(X) given by j(P) = {Z(f) : f ∈ P} (see [14, Chapter 2]). If P ∈ PZ(X) and P ⊆ Mx for some x ∈ X then Ox ⊆ P [14, 4I], and hence Hx ⊆ LZ[P ] ⊆ ˜ Jx by Proposition 2.5. Every z-ideal of CR(X) is an intersection of prime z-ideals and the minimal prime ideals of CR(X) are z-ideals [14, 2.8, 14.7]. The prime ideals containing a given prime ideal form a chain [14, 14.8]. Example. Let X = N ∪ {ω} be the one-point compactification of N and set A = C(X) ⊗ K(H). Then Mx = Ox for x ∈ N, but Mω = Oω. The assignment F → PF = {f ∈ CR(X) : Z(f) {ω} ∈ F} gives a bijection between the family of free ultrafilters on N (every ultrafilter on N is trivially a z-ultrafilter) and the family of non-maximal prime z-ideals contained in Mω. Each PF is a minimal prime z-ideal [14, 14G] and we shall see in Section 4 that its image LF under the mapping of Theorem 2.4 is a minimal closed primal ideal of M(A). The ideal Hω = LZ(Oω) is a Glimm ideal but is not primal. 3. The homeomorphism onto MinPrimal(M(A)) In this section we specialize to the case when A is a σ-unital quasi-standard C∗ - algebra. We will be assuming that A is canonically represented as a C0(X)-algebra with the base map φ as the complete regularization map for Prim(A) and with X = Xφ = Glimm(A). For the main result we will also need to assume that A/Jx is non-unital for x ∈ X (note that this is automatically satisfied if A is stable).
- 37. Minimal Primal Ideals in the Multiplier Algebra 23 The reasons for restricting to quasi-standard C∗ -algebras are twofold. The first is the fact, already mentioned, that when A is quasi-standard, MinPrimal(A) and Glimm(A) coincide as sets (and indeed as topological spaces). This has the implication that, for x ∈ X = Glimm(A), the ideal ˜ Jx is primal in M(A) [6, Lemma 4.5]; and hence there must be minimal closed primal ideals of M(A) lying between ˜ Jx and the Glimm ideal Hx of M(A). But secondly, if A is quasi-standard then norm functions of elements of A are continuous on Glimm(A), so for a ∈ A, Z(a) is a zero set of Glimm(A). Furthermore if A is also σ-unital and u ∈ A is a strictly positive element then, as we have already mentioned, for b ∈ M(A) Z(b) = Z(bu), so Z(b) is also a zero set of Glimm(A). Thus the elaborate machinery of zero sets works smoothly for this class of algebras. For a ring R let Min(R) be the space of minimal (algebraic) primal ideals of R with the lower topology generated by sub-basic sets of the form {P ∈ Min(R) : a / ∈ P} as a varies through elements of R. If R is a commutative ring then an argument of Krull shows that every minimal primal ideal of R is prime, and Min(R) is the usual space of minimal prime ideals of R with the hull-kernel topology, see [28] and the references given there. If P is a minimal prime ideal of CR(X) then P is a z-ideal, as we have mentioned, so an obvious step is to identify the image of Min(CR(X)) under the embedding map L of Theorem 2.4. We shall show that the embedding map L carries Min(CR(X)) homeomorphically onto MinPrimal(M(A)) with the τw-topology (where the τw-topology is defined on MinPrimal(A) by taking sets of the form {P ∈ MinPrimal(A) : a / ∈ P} (a ∈ A) as sub-basic; see [2, p. 525] where an equivalent definition is given). It is convenient to proceed in two stages. In Theorem 3.2 we show that the assignment P → Lalg Z[P ] defines a homeomorphism Θ from Min(CR(X)) onto Min(M(A)). For this theorem we do not require the quotients A/Jx (x ∈ X) to be non-unital. Then in Theorem 3.4 we show that, if these quotients are non-unital, the assignment Lalg F → LF defines a homeomorphism Φ from Min(M(A)) onto MinPrimal(M(A)). The method of proof of Theorem 3.2 is similar to that of [28, Theorem 3.2] except that we are here working with filters of zero sets rather than with ideals of cozero sets. For further work on the space of minimal (algebraic) primal ideals of a C∗ -algebra, see [29] and [30]. For a C∗ -algebra B and a ∈ B, let Ia be the closed ideal of B generated by a. The following lemma is a special case of [28, Theorem 2.3], which itself is a special case of a more general result due to Keimel [18]. Recall that ideals are not necessarily closed unless stated to be so. Lemma 3.1. Let B be a C∗ -algebra and let P be a primal ideal of B. Then P is a minimal primal ideal if and only if for all a ∈ P there exist b1, . . . , bn ∈ B P such that IaIb1 . . . Ibn = {0}. Let I⊥ a be the largest ideal of B such that IaI⊥ a = {0}. Then Lemma 3.1 implies that if P is a minimal primal ideal of B and a ∈ P then I⊥⊥ a ⊆ P.
- 38. 24 R.J. Archbold and D.W.B. Somerset Theorem 3.2. Let A be a σ-unital quasi-standard C∗ -algebra and set X=Glimm(A). Then the assignment P → Lalg Z[P ] defines a homeomorphism Θ from Min(CR(X)) onto Min(M(A)). Proof. First we show that if F = Z[P] for P ∈ Min(CR(X)) then Lalg F is a minimal primal ideal of M(A). Let bi ∈ M(A) Lalg F (1 ≤ i ≤ n). Then Z(bi) / ∈ F for each i, so Z(b1)∪· · ·∪Z(bn) / ∈ F since F is a prime z-filter. Hence Z(b1)∪· · ·∪Z(bn) = X, so there exists x ∈ X such that bi / ∈ ˜ Jx (1 ≤ i ≤ n). Since ˜ Jx is primal, b1M(A) . . . M(A)bn = {0}. Hence Lalg F is primal. Now let b ∈ Lalg F with b = 0. Then Z(b) ∈ F, so by [19, Lemma 3.1] there exists f ∈ CR(X) such that Z(f)∪Z(b) = X and Z(f) / ∈ F. Let c ∈ A with Z(c) = Z(f). Then Z(c) / ∈ F so c / ∈ Lalg F , and Z(c) ∪ Z(b) = X, so bM(A)c = {0} by Proposition 2.3(ii). This shows that Lalg F is a minimal primal ideal of M(A) and hence that Θ maps into Min(M(A)). Now let P and Q be distinct elements of Min(CR(X)). Then Z[P] = Z[Q], and since for each zero set Z there exists c ∈ A with Z(c) = Z, it follows that Lalg Z[P ] = Lalg Z[Q]. This shows that Θ is injective. Now suppose that Q ∈ Min(M(A)) and let G = {Z(b) : b ∈ Q}. We show that G is a minimal prime z-filter on X. First note that if b ∈ Q then I⊥ b is non-zero by Lemma 3.1, and indeed I⊥ b = {a ∈ M(A) : Z(a) ∪ Z(b) = X} by the primality of the ideals ˜ Jx (x ∈ X). Hence Z(b) is non-empty, so ∅ / ∈ G. For b, c ∈ Q, Z(b) ∩ Z(c) = Z(bb∗ + cc∗ ) ∈ G. If b ∈ Q and c ∈ M(A) with Z(c) ⊇ Z(b) then Z(a) ∪ Z(c) = X for all a ∈ I⊥ b , so c ∈ I⊥⊥ b ⊆ Q, as observed after Lemma 3.1. Hence Z(c) ∈ G. This shows that G is a proper z-filter, and also that Q = Lalg G . To show that G is a prime z-filter, let Z1 and Z2 be zero sets of X such that Z1 ∪ Z2 = X. Let b, c ∈ A such that Z1 = Z(b) and Z2 = Z(c). Then bM(A)c = {0}, so at least one of b and c (b say) belongs to Q since Q is primal. Hence Z1 ∈ G. This shows that G is prime [14, 2E]. To see that G is minimal prime, let Z ∈ G and let b ∈ Q such that Z(b) = Z. Then by Lemma 3.1 there exist c1, . . . , cn ∈ M(A)Q such that IbIc1 . . . Icn = {0}. Hence Z(ci) / ∈ G (1 ≤ i ≤ n), by an argument in the previous paragraph, and Z(b) ∪ Z(c1) ∪ · · · ∪ Z(cn) = X by the primality of the ideals ˜ Jx (x ∈ X). Set Y = Z(c1) ∪ · · · ∪ Z(cn). Then Y is a zero set in X, being a finite union of zero sets, and Y / ∈ G since G is prime. Since Z ∪Y = X it follows that no z-filter strictly smaller than G can be prime. Hence G is a minimal prime z-filter, and Q = Lalg G belongs to the range of Θ. Thus Θ is a bijection. Finally, for f ∈ CR(X) we can find a ∈ A such that Z(a) = Z(f); and conversely, given a ∈ M(A), since A is σ-unital and quasi-standard we can find f ∈ CR(X) such that Z(a) = Z(f). Hence in either case Θ({P ∈ Min(CR(X)) : f / ∈ P}) = Θ({P ∈ Min(CR(X)) : Z(f) / ∈ Z[P]}) = {Lalg Z[P ] ∈ Min(M(A)) : Z(a) / ∈ Z[P]} = {Lalg Z[P ] ∈ Min(M(A)) : a / ∈ Lalg Z[P ]}.
- 39. Minimal Primal Ideals in the Multiplier Algebra 25 Since the hull-kernel topology on Min(CR(X)) can be defined either using ideals or using elements, it follows that Θ is a homeomorphism. A comparison of the proof of Theorem 3.2 with that of [28, Theorem 3.2] shows that when A is a σ-unital quasi-standard C∗ -algebra, the assignment Q → Q ∩ A gives a homeomorphism from Min(M(A)) onto Min(A). For the next theorem, we need the following family of functions which is useful for relating LF and Lalg F . For 0 1/2, define the continuous piecewise linear function f : [0, ∞) → [0, ∞) by: (i) f (x) = 0 (0 ≤ x ≤ ); (ii) f (x) = 2(x − ) ( ≤ x ≤ 2 ); (iii) f (x) = x (2 ≤ x). Note that for b ∈ M(A)+ , if b ∈ LF then f (b) belongs to the Pedersen ideal of LF for all [24, 5.6.1], and hence f (b) ∈ Lalg F . On the other hand, b − f (b) ≤ . Thus we have the following lemma. Lemma 3.3. Let A be C0(X)-algebra with base map φ and let F be a z-filter on Xφ. Let b ∈ M(A)+ . Then with the notation above, b ∈ LF if and only if f (b) ∈ Lalg F for all ∈ (0, 1/2). Theorem 3.4. Let A be a σ-unital, quasi-standard C∗ -algebra with A/G non-unital for all G ∈ Glimm(A) and set X = Glimm(A). Then the assignment P → LZ[P ] defines a homeomorphism from Min(CR(X)) onto MinPrimal(M(A)). Proof. By Theorem 3.2, it is enough to show that the assignment Lalg Z[P ] → LZ[P ] (P ∈ Min(CR(X))) defines a homeomorphism Φ from Min(M(A)) onto MinPrimal(M(A)). If R is a minimal closed primal ideal of M(A) then R contains some Lalg Z[P ] ∈ Min(M(A)), and hence R = LZ[P ]. Thus the range of Φ certainly contains MinPrimal(M(A)). Furthermore, Theorem 2.4 implies that Φ is injective and also that if P, Q ∈ Min(CR(X)) with P = Q then LZ[P ] ⊆ LZ[Q]. Suppose that Q ∈ Min(CR(X)). Then Lalg Z[Q] ∈ Min(M(A)) so LZ[Q] is a closed primal ideal of M(A). Hence LZ[Q] contains a minimal closed primal ideal of M(A), which we have just seen is of the form LZ[P ] for P ∈ Min(CR(X)). Thus P = Q, so the range of Φ equals MinPrimal(M(A)). Hence Φ is a bijection. Now let a ∈ M(A)+ and let Z = Z(a), a zero set in X. Then by [7, Corollary 3.1] there exists c ∈ M(A)+ such that c + ˜ Jx = 1 for x ∈ X Z and c ∈ ˜ Jx for x ∈ Z. Hence Z(f (c)) = Z for all ∈ (0, 1/2). Thus Φ({Lalg F ∈ Min(M(A)) : a / ∈ Lalg F }) = Φ({Lalg F ∈ Min(M(A)) : Z / ∈ F}) = {LF ∈ MinPrimal(M(A)) : c / ∈ LF }, by Lemma 3.3. On the other hand, again by Lemma 3.3, Φ−1 ({LF ∈ MinPrimal(M(A)) : a / ∈ LF }) = ∈(0,1/2) {Lalg F ∈ Min(M(A)) : f (a) / ∈ Lalg F }). Thus it follows that Φ is a homeomorphism.
- 40. 26 R.J. Archbold and D.W.B. Somerset Corollary 3.5. Let A be a σ-unital, continuous C0(X)-algebra with base map φ such that A/Jx is non-unital and φ−1 (x) is contained in a limit set in Prim(A) for all x ∈ Xφ. Then Min(CR(Xφ)) is homeomorphic to MinPrimal(M(A)). Proof. By Lemma 2.1, A is quasi-standard and there is a homeomorphic map ψ : Glimm(A) → Xφ. For G ∈ Glimm(A), there exists x ∈ Xφ such that ψ−1 (x) = G. Hence Jx = G, so A/G is non-unital. The result now follows from Theorem 3.4. 4. Applications The space of minimal prime ideals of CR(X) has been studied in numerous papers, e.g., [19], [15], [12], [11], [16], so Theorem 3.4 has various immediate corollaries. We present a sample of these. Recall that a topological space Y is countably compact if every countable open cover of Y has a finite subcover. If Y is a T1-space then Y is countably compact if and only if every infinite subset of Y has a limit point in Y [23, p. 181]. Corollary 4.1. Let A be a σ-unital, quasi-standard C∗ -algebra with A/G non-unital for all G ∈ Glimm(A). (i) The Hausdorff space MinPrimal(M(A)) is totally disconnected and countably compact. (ii) If MinPrimal(M(A)) is locally compact then it is basically disconnected. Proof. (i) The space of minimal closed primal ideals of a C∗ -algebra is always Hausdorff in the τw-topology [2, Corollary 4.3]. The total disconnectedness and countable compactness follow from Theorem 3.4 and from [15, Corollary 2.4] and [15, Theorem 4.9] respectively. (ii) This follows from Theorem 3.4 and [15, Theorem 4.7]. In the context of Corollary 4.1, recall that a necessary and sufficient condi- tion for M(A) to be quasi-standard is that Glimm(M(A)) and MinPrimal(M(A)) should coincide both as sets and as topological spaces [5, Theorem 3.3]. Since M(A) is unital, Glimm(M(A)) is compact, so MinPrimal(M(A)) would also have to be compact. By Corollary 4.1(ii), this implies that MinPrimal(M(A)), and hence Glimm(M(A)), would have to be basically disconnected; and this in turn implies that Glimm(A) would have to be basically disconnected [14, 6M.1]. Thus we recover the necessity of Glimm(A) being basically disconnected if M(A) is quasi-standard. In point of fact, it was shown in [6, Corollary 4.9] that if A is a σ-unital quasi- standard C∗ -algebra with centre equal to {0} then M(A) is quasi-standard if and only if Glimm(A) is basically disconnected. Corollary 4.2. Let A be a σ-unital, quasi-standard C∗ -algebra and suppose that A/G is non-unital for all G ∈ Glimm(A). Then the following are equivalent: (i) MinPrimal(M(A)) is compact;
- 41. Minimal Primal Ideals in the Multiplier Algebra 27 (ii) Glimm(A) is cozero-complemented; that is, for every cozero set U in Glimm(A) there exists a cozero set V in Glimm(A) such that U ∩ V = ∅ and U ∪ V is dense in Glimm(A). Proof. This follows by Theorem 3.4 and the characterization in [15, Corollary 5.5]. For example, if Glimm(A) is basically disconnected or is homeomorphic to an ordi- nal space then Glimm(A) is cozero complemented [16, Examples 1.6], so the space MinPrimal(M(A)) is compact. On the other hand, if Glimm(A) is the Alexandroff double of a compact metric space without isolated points then Glimm(A) is com- pact and first countable but not cozero complemented [16, Examples 1.7]. Hence MinPrimal(M(A)) is not compact. If A is separable, much more can be said. Recall that a regular closed set is one that is the closure of its interior. If A is separable then Glimm(A) is perfectly normal [8, Lemma 3.9] (i.e., every closed subset of Glimm(A) is a zero set) so A certainly satisfies condition (ii) of the next corollary. Corollary 4.3. Let A be a σ-unital, quasi-standard C∗ -algebra. Suppose that A/G is non-unital for G ∈ Glimm(A). Then the following are equivalent: (i) MinPrimal(M(A)) is compact and extremally disconnected; (ii) Every regular closed set in Glimm(A) is the closure of a cozero set. In particular, if A is separable then A satisfies these equivalent conditions. Proof. This follows by Theorem 3.4 and the characterization in [15, Theorems 4.4 and 5.6]. More generally, recall that a topological space X has the countable chain con- dition if every family of non-empty pairwise disjoint open subsets of X is countable. It is easily seen that a completely regular topological space with the countable chain condition has property (ii) of Corollary 4.3. If a C∗ -algebra A has a faithful representation on a separable Hilbert space, then Glimm(A) satisfies the countable chain condition [30, p. 85]. We conclude with one further application of Theorem 3.4. Corollary 4.4. Set A = C(βN N) ⊗ K(H). Then MinPrimal(M(A)) is nowhere locally compact. If Martin’s Axiom holds then MinPrimal(M(A)) is not an F-space. Proof. Both statements follow from Theorem 3.4, the first by [15, Example 5.9], and the second by [12, Corollary 4].
- 42. 28 R.J. Archbold and D.W.B. Somerset References [1] C.A. Akemann, G.K. Pedersen and J. Tomiyama, Multipliers of C∗ -algebras. J. Funct. Anal. 13 (1973), 277–301. [2] R.J. Archbold, Topologies for primal ideals. J. London Math. Soc. (2) 36 (1987), 524–542. [3] R.J. Archbold and C.J.K. Batty, On factorial states of operator algebras, III. J. Operator Theory 15 (1986), 53–81. [4] R.J. Archbold, E. Kaniuth and D.W.B. Somerset, Norms of inner derivations for multipliers of C∗ -algebras and group C∗ -algebras. J. Functional Analysis 262 (2012), 2050–2073. [5] R.J. Archbold and D.W.B. Somerset, Quasi-standard C∗ -algebras. Math. Proc. Camb. Phil. Soc 107 (1990), 349–360. [6] R.J. Archbold and D.W.B. Somerset, Multiplier algebras of C0(X)-algebras. Münster J. Math. 4 (2011), 73–100. [7] R.J. Archbold and D.W.B. Somerset, Ideals in the multiplier and corona algebras of a C0(X)-algebra. J. London Math. Soc. (2) 85 (2012), 365–381. [8] R.J. Archbold and D.W.B. Somerset, Spectral synthesis in the multiplier algebra of a C0(X)-algebra. Quart. J. Math. Oxford 65 (2014), 1–24. [9] R.J. Archbold and D.W.B. Somerset, Separation properties in the primitive ideal space of a multiplier algebra. Israel J. Math. 200 (2014), 389–418. [10] R.C. Busby, Double centralizers and extensions of C∗ -algebras. Trans. Amer. Math. Soc. 132 (1968), 79–99. [11] A. Dow, The space of minimal prime ideals of C(βN N) is probably not basically disconnected, pp. 81–86, General Topology and Applications, (Lecture Notes in Pure and Applied Math., 123), Dekker, NY, 1990. [12] A. Dow, M. Henriksen, R. Kopperman and J. Vermeer, The space of minimal prime ideals of C(X) need not be basically disconnected. Proc. Amer. Math. Soc. 104 (1988), 317–320. [13] G.A. Elliott, Derivations of matroid C∗ -algebras, II. Ann. Math.(2) 100 (1974), 407– 422. [14] L. Gillman and M. Jerison, Rings of Continuous Functions. Van Nostrand, New Jersey, 1960. [15] M. Henriksen and M. Jerison, The space of minimal prime ideals of a commutative ring. Trans. Amer. Math. Soc. 115 (1965), 110–130. [16] M. Henriksen and R.G. Woods, Cozero complemented space; when the space of min- imal prime ideals of a C(X) is compact. Topology and Its Applications 141 (2004), 147–170. [17] E. Kaniuth, G. Schlichting and K. Taylor, Minimal primal and Glimm ideal spaces of group C∗ -algebras. J. Funct. Anal. 130 (1995), 45–76. [18] K. Keimel, A unified theory of minimal prime ideals. Acta Math. Acad. Sci. Hun- garicae 23 (1972), 51–69. [19] J. Kist, Minimal prime ideals in commutative semigroups. Proc. London. Math. Soc. (3) 13 (1963), 31–50.
- 43. Minimal Primal Ideals in the Multiplier Algebra 29 [20] D. Kucerovsky and P.W. Ng, Nonregular ideals in the multiplier algebra of a stable C∗ -algebra. Houston J. Math. 33 (2007), 1117–1130. [21] H.X. Lin, Ideals of multiplier algebras of simple AF-algebras. Proc. Amer. Math. Soc. 104 (1988), 239–244. [22] M. Mandelker, Prime z-ideal structure of C(R). Fund. Math. 63 (1968), 145–166. [23] J.R. Munkres, Topology. 2nd Edition, Prentice-Hall, New Jersey, 1999. [24] G.K. Pedersen, C∗ -algebras and their Automorphism Groups. Academic Press, Lon- don, 1979. [25] F. Perera, Ideal structure of multiplier algebras of simple C∗ -algebras with real rank zero. Can. J. Math. 53 (2001), 592–630. [26] H.L. Pham, Uncountable families of prime z-ideals in C0(R). Bull. London Math. Soc. 41 (2009), 354–366. [27] M. Rørdam, Ideals in the multiplier algebra of a stable C∗ -algebra. J. Operator Th. 25 (1991), 283–298. [28] D.W.B. Somerset, Minimal primal ideals in Banach algebras. Math. Proc. Camb. Phil. Soc. 115 (1994), 39–52. [29] D.W.B. Somerset, The local multiplier algebra of a C∗ -algebra. Quart. J. Math. Oxford (2) 47 (1996), 123–132. [30] D.W.B. Somerset, Minimal primal ideals in rings and Banach algebras. J. Pure and Applied Algebra 144 (1999), 67–89. [31] D.P. Williams, Crossed Products of C∗ -algebras. American Mathematical Society, Rhode Island, 2007. R.J. Archbold and D.W.B. Somerset Institute of Mathematics University of Aberdeen Kings College Aberdeen AB24 3UE Scotland, UK e-mail: r.archbold@abdn.ac.uk dwbsomerset@gmail.com
- 44. Operator Theory: Advances and Applications, Vol. 250, 31–48 Countable Spectrum, Transfinite Induction and Stability Wolfgang Arendt Dedicated to Charles Batty on the occasion of his sixtieth birthday Abstract. We reconsider the contour argument and proof by transfinite in- duction of the ABLV-Theorem given in [AB88]. But here we use the method to prove a Tauberian Theorem for Laplace transforms which has the ABVL- Theorem about stability of a semigroup as corollary and also gives quantita- tive estimates. It is interesting that considering countable spectrum leads to the same problems Cantor encountered when he tried to prove a uniqueness result for trigonometric series. It led him to invent ordinal numbers and transfinite induction. We explain these connections in the article. Mathematics Subject Classification (2010). 47D06, 44A10. Keywords. Semigroups, asymptotic behavior, Laplace transform, Tauberian theorem, countable spectrum, transfinite induction, uniqueness theorem for trigonometric series. 1. Introduction Frequently, it is worthwhile to revisit a mathematical result with the benefit of several years’ hindsight. Things may appear in a different light, different methods might be known. The result I am talking about here is the stability theorem I proved together with Charles Batty 25 years ago, which says the following. Let (T (t))t≥0 be a bounded C0-semigroup with generator A. If σ(A) ∩ iR is countable and σp(A ) ∩ iR = ∅ (where σp(A ) denotes the point spectrum of the adjoint), then the semigroup is stable; i.e., limt→∞ T (t)x = 0 for all x ∈ X. It was not necessary to wait 25 years for new methods to appear. In fact, ex- actly at the same time, this stability result was obtained independently by Ljubich and Vu [LV88] at the University of Kharkov in the Soviet Union by completely dif- ferent methods. For this reason the result is frequently called the ABLV-Theorem. Ljubich and Vu use a quotient method, and later Fourier methods were developed by Esterle, Strouse and Zouakia [ESZ92] (spectral synthesis) and it was Chill c Springer International Publishing Switzerland 2015
- 45. 32 W. Arendt [Chi98] who shed new light onto the old methods of Ingham – 80 years after their first appearance. These different approaches all have their advantages and mer- its. The proof by Ljubich and Vu is the most functional analytical in nature. Its disadvantage is that it merely works in the context of semigroups and not more generally for Laplace transforms. The advantage of Chill’s approach is that it is valid for Laplace transforms (see [Chi98], [ABHN11, Theorem 4.9.7]). We refer to the survey by Chill and Tomilov [CT07] for more information and also to Section 5.5 of [ABHN11]. Still, we want to revisit our proof from 1986, which used two ingredients, a contour argument and transfinite induction. Compared with the other methods, there are two advantages: our proof is completely elementary and it also gives quan- titative results (which have grown in importance recently, see Batty [Bat90], Batty and Duykaerts [BD08], Borichev and Tomilov [BT10], Batty, Chill and Tomilov [BCT], [BBT14] as well as Section 4.4 in [ABHN11]). Concerning elegance and esthetics, the opinions of colleagues are not unan- imous. Most people believe that our method is quite technical and even we did not use it in our book [ABHN11] to prove the ABLV-Theorem. Still, we believe that the transfinite induction argument we used in 1986 is quite striking and even elegant. Once the inductive statement is formulated in the right way, its proof is automatic. Our aim in this article is to make this transparent by formulating the technical part in an abstract and easy way (Lemma 3.6). But we also arrange the arguments differently and obtain a new interesting result, namely a (quantitative) Tauberian theorem for Laplace transforms (Theorem 3.1) where an exceptional countable set occurs in the hypothesis. It is this result which we prove by transfi- nite induction in the present article (in contrast to our original proof [AB88] where the argument by transfinite induction was done on the level of the semigroup). The powerful Mittag-Leffler Theorem, a topological argument in the spirit of Baire’s theorem, allows one to pass from the Tauberian theorem to the ABLV-Theorem, see Section 4. Our Tauberian theorem gives also an improvement of a Tauberian theorem for power series by Allan, O’Farrell and Randsford [AOR87] which was motivated by the Katznelson–Tzafriri theorem. Concerning the contour estimates, they demonstrate the power of Cauchy’s Theorem and are most elegant when the spectrum on the imaginary axis is empty. As an appetizer we consider this case in Section 2 emphasizing the quantitative character. In Section 3 we prove the general Tauberian theorem elaborating the use of transfinite induction. It is interesting that Cantor encountered similar problems as we did in the context of countable spectrum when he tried to prove a uniqueness result for trigonometric series where a closed, countable exceptional set has to be mastered. It was this problem which led him to develop set theory, ordinal numbers and transfinite induction. In Section 5 we take the opportunity to present the solution of Cantor’s problem by transfinite induction, a striking resemblance to our proof, a resemblance of which we were not aware in 1986. Cantor must have been aware of the argument, but he never published the end of the proof.
- 46. Countable Spectrum, Transfinite Induction and Stability 33 2. Empty spectrum This section is an introduction to the subject where we consider the simplest case of a complex Tauberian theorem, the Newman-trick for contour integrals and the special case of the ABLV-Theorem where the spectrum on the imaginary axis is empty. The results are contained in [AB88], [AP92] (see also [ABHN11]). Here however, we put them together in a way which makes transparent the quantitative nature of the results and which demonstrates the power of the contour argument in a simple case. The more refined techniques are then presented in Section 3. We consider a function f ∈ L∞ (R+, X) where X is a complex Banach space, R+ = [0, ∞). By ˆ f(λ) := ∞ 0 e−λt f(t) dt (Re λ 0) we denote the Laplace transform of f. It is a holomorphic function defined on the right half-plane C+. If F(t) := t 0 f(s) ds converges to F∞ as t → ∞, then limλ→0 ˆ f(λ) = F∞. This Abelian theorem is easy to see. The converse is false in general: If limλ→∞ ˆ f(λ) = F∞ exists, then t 0 f(s) ds need not converge as t → ∞. But if a theorem says that it does under some additional hypothesis then we call it a Tauberian theorem and the additional hypothesis a Tauberian condition. An interesting Tauberian theorem is the following. Theorem 2.1 (Newman–Korevaar–Zagier). Assume that ˆ f has a holomorphic ex- tension to an open set containing C+. Then lim t→∞ t 0 f(s) ds exists. It follows that limt→∞ t 0 f(s) ds = ˆ f(0) by the remark above. Here the Tauberian condition is that ˆ f can be extended to a holomorphic function on an open set containing C+. A theorem of this type had already been proved by Ing- ham [Ing35] in the thirties (see also Korevaar’s book [Kor04, p. 135]). But Newman [New80] found an elegant contour argument (which he applied to Dirichlet series), that was used by Korevaar [Kor82] and Zagier [Zag97] for Laplace transforms to give beautiful proofs of the prime number theorem. Here is an estimate, which implies Theorem 2.1 and which shows the simplicity of the argument as well as its quantitative aspect. We let f∞ := supt≥0 f(t). Proposition 2.2. Let R 0. Assume that ˆ f has a holomorphic extension to a neighborhood of C+ ∪ i[−R, R]. Then lim sup t→∞ t 0 f(s) − ˆ f(0) ≤ f∞ R .
- 47. 34 W. Arendt Proof. Let g = ˆ f and for t 0 let gt(z) = t 0 e−zs f(s) ds. Thus gt is an entire function. Let U be an open, simply connected set containing i[−R, R]∪C+. Denote by γ a path going from iR to −iR lying entirely in U ∩{z ∈ C : Re z 0} besides the endpoints. We apply Cauchy’s Theorem to this contour. The introduction of an ad- ditional fudge factor under the following integral is the ingenious trick due to Newman. t 0 f(s) ds − ˆ f(0) = gt(0) − g(0) = 1 2πi |z|=R Re z0 (gt(z) − g(z))etz 1 + z2 R2 dz z + 1 2πi γ (gt(z) − g(z))etz 1 + z2 R2 dz z =: I1(t) + I2(t). It follows from the Dominated Convergence Theorem that limt→∞ I2(t) = 0. In order to estimate I1(t) let z = Reiθ , |θ| π 2 , be on the right-hand semi- circle. Then on the one hand (gt(z) − g(z))etz = ∞ t e−zs f(s) ds etz ≤ f∞ ∞ t e−sR cos θ ds etR cos θ ≤ f∞ R cos θ and on the other |1 + z2 R2 | = |1 + ei2θ | = |e−iθ + eiθ | = 2 cos θ. Thus I1(t) ≤ 1 2π π f∞ R cos θ 2 cosθ = f∞ R and the proposition is proved. In 1986 when we worked in Oxford on stability of semigroups we knew a version of Theorem 2.1 from an unpublished manuscript by Zagier (cf. [Zag97]). It was easy to apply it to semigroups: Let (T (t))t≥0 be a C0-semigroup with generator A. Assume that T (t) ≤ M for all t ≥ 0. For x ∈ X let f(t) = T (t)x. Then ˆ f(λ) = R(λ, A)x. Now assume that
- 48. Countable Spectrum, Transfinite Induction and Stability 35 σ(A) ∩ iR = ∅. Then ˆ f(0) = −A−1 x and the Newman–Korevaar–Zagier Theorem 2.1 implies that t 0 f(s) ds = t 0 T (s)AA−1 x ds = T (t)A−1 x − A−1 x converges to −A−1 x as t → ∞. Hence T (t)A−1 x → 0 as t → ∞ for all x ∈ X. Since rg A−1 = D(A) is dense in X and T (t) ≤ M it follows that limt→∞ T (t)x = 0 for all x ∈ X; i.e., the semigroup is stable. We have proved the following. Theorem 2.3. Assume that (T (t))t≥0 is a bounded C0-semigroup with generator A. If σ(A) ∩ iR = ∅, then limt→∞ T (t)x = 0 for all x ∈ X. It is natural to ask what happens if σ(A) ∩ iR = ∅. If iη ∈ σp(A ), the point spectrum of the adjoint A of A, then T (t) x = eiηt x for all t ≥ 0 and some x ∈ X {0}. Let x ∈ X be such that x , x = 1. Then T (t)x, x = eiηt for all t ≥ 0 and so the semigroup is definitely not stable. Thus σp(A ) ∩ iR = ∅ (2.1) is a necessary condition for stability. By the Hahn–Banach Theorem, condition (2.1) is equivalent to rg(iη − A) being dense in X (2.2) where rg stands for the range of the operator. What is special for x ∈ rg(iη − A)? Let x = (iη − A)y where y ∈ D(A), f(t) = T (t)x as before. Then t 0 f(s)e−iηs ds = y − e−iηt T (t)y. Thus sup t≥0 t 0 f(s)e−iηs ds ∞. (2.3) This condition turns out to be useful for proving a Tauberian theorem by the contour method if iη is a singular point. Before discussing this in the next section we point out a generalization of the Tauberian Theorem 2.1. It is not necessary to assume that a holomorphic extension exists, a continuous extension suffices. Theorem 2.4. Let f ∈ L∞ (R+, X), R 0, F∞ ∈ X. Assume that 1 λ ( ˆ f(λ) − F∞) has a continuous extension to C+ ∪ i[−R, R]. Then lim sup t→∞ t 0 f(s) ds − F∞ ≤ 2f∞ R . (2.4)
- 49. 36 W. Arendt This is obtained by a modification of the contour argument above (cf. [AP92, Lemma 5.2], where a more complicated situation is considered). We give the proof of Theorem 2.4 in order to be complete. It is interesting that now, instead of the Dominated Convergence Theorem, we use the Riemann–Lebesgue Theorem for Fourier coefficients. The price is a factor 2 appearing in the estimate (2.4) in contrast to the better estimate given in Proposition 2.2. Proof. First case: F∞ = 0. Let g = ˆ f. Thus g(z) z has a continuous extension to C+ ∪ i[−R, R]. By (a slight extension of) Cauchy’s Theorem one has γ g(z) z (1 + z2 R2 )etz dz + |z|=R Re z0 g(z) z (1 + z2 R2 )etz dz = 0 (2.5) where γ is the straight line from iR to −iR. For t 0 consider the entire function gt(z) = t 0 e−sz f(s) ds. Thus by (2.5), t 0 f(s) ds = 1 2πi |z|=R gt(z) 1 + z2 R2 etz dz z = 1 2πi |z|=R Re z0 (gt(z) − g(z)) 1 + z2 R2 etz dz z − 1 2πi γ g(z)etz 1 + z2 R2 dz z + 1 2πi |z|=R Re z0 gt(z) 1 + z2 R2 etz dz z =: I1(t) + I2(t) + I3(t). By the Riemann–Lebesgue Theorem, limt→∞ I2(t) = 0. One has I1(t) ≤ 1 R f∞ for all t ≥ 0 as in Proposition 2.2. The integral I3(t) can be estimated in a similar way, lim sup t→∞ I3(t) ≤ 1 R f∞. Thus lim supt→∞ t 0 f(s) ds ≤ 2 R f∞. Second case: F∞ ∈ X is arbitrary. Let ϕ: [0,∞) → R be continuous with compact support satisfying 1 0 ϕ(s)ds = 1. Let f1(t) := f(t) − ϕ(t)F∞. Then ˆ f1(λ) = ˆ f(λ) − ϕ̂(λ)F∞, ϕ̂(0) = 1. Thus ˆ f1(λ) λ = ˆ f(λ) − F∞ λ − ϕ̂(λ) − ϕ̂(0) λ F∞ has a continuous extension to C+ ∪ i[−R, R].
- 50. Countable Spectrum, Transfinite Induction and Stability 37 By the first case lim sup t→∞ t 0 f(s) ds − F∞ ≤ 2 R f∞. Applying the preceding results to f(· + s) instead of f one even obtains the estimate lim sup t→∞ t 0 f(s) ds − F∞ ≤ 1 R lim sup t→∞ f(t), (2.6) in Proposition 2.2 and the estimate lim sup t→∞ t 0 f(s) ds − F∞ ≤ 2 R lim sup t→∞ f(t), (2.7) instead of (2.4), cf. [AP92, Remark 9.2]. We finish this section by going back to the origins of Tauberian theory. Given a bounded sequence (an)n∈N0 consider the power series p(z) = ∞ n=0 anzn which is defined for |z| 1. If ∞ n=0 an =: b∞ exists then Abel showed in 1826 that lim x 1 p(x) = b∞. (2.8) The converse is not true in general. Additional assumptions are needed. It was Tauber who proved in 1897 that the series converges if in addition to (2.8) one assumes that lim n→∞ nan = 0, (2.9) thus proving the first “Tauberian theorem”. Littlewood showed in 1911 that the “Tauberian condition” (2.9) can be re- laxed to supn∈N nan ∞. Another Tauberian theorem is due to Riesz. It is actually a consequence of the estimate (2.6). Theorem 2.5 (Riesz). Let an ∈ X, n ∈ N0, such that limn→∞ an = 0. Assume that the power series p(z) = ∞ n=0 anzn (|z| 1) has a holomorphic extension to an open neighborhood of 1. Then ∞ n=0 an = p(1). Proof. Let f(t) = an if t ∈ [n, n + 1). Then f ∈ L∞ (R+, X) and ˆ f(λ) = 1 − e−λ λ p(e−λ ) (Re λ 0).
- 51. 38 W. Arendt Thus ˆ f has a holomorphic extension to a disc of radius 2R centered at 0 for some R 0 and ˆ f = p(1). Thus (2.6) implies that lim n→∞ n k=0 ak − p(1) ≤ 1 R lim sup n→∞ an = 0. This proof is taken from [AP92, Remark 3.4]. 3. A complex Tauberian theorem Let f ∈ L∞ (R+, X). The Laplace transform ˆ f of f is a holomorphic function from the open right-hand half-plane C+ into X. If F(t) := t 0 f(s) ds converges to F∞ as t → ∞, then limλ 0 ˆ f(λ) = F∞ by an easy Abelian theorem. As in Section 2, we want to prove the converse. But here we will relax the assumptions considerably. As in Theorem 2.4 we will estimate lim sup t→∞ F(t) − F∞. The Tauberian condition is expressed in terms of the boundary behavior of ˆ f(λ) as λ → iη. Theorem 3.1. Let R 0, F∞ ∈ X. Let E ⊂ (−R, 0) ∪ (0, R) be closed and countable. Assume that (a) ˆ f(λ)−F∞ λ has a continuous extension to C+ ∪ i([−R, R] E) and that (b) supt≥0 t 0 e−iηs f(s) ds ∞ for all η ∈ E. Then lim sup t→∞ t 0 f(s) ds − F∞ ≤ 2f∞ R . (3.1) Our point is that the bound in (b) may depend on η ∈ E. In the case where it is independent, (3.1) can be proved purely by a contour argument (see [AP92, Theorem 3.1], and [AB88, Theorem 4.1] for a slightly more special case). Since we do not assume a uniform bound in (b) our proof needs an argument of trans- finite induction. It is similar to the transfinite induction argument given for the proof of the ABLV-Theorem in [AB88] and, we think, an interesting mathematical argument in its own right. Here it is. As in the proof of Proposition 2.4 we may assume that F∞ = 0 which we do now. We assume the hypotheses of Theorem 3.1. For the proof we denote by Jn the set of all (η1, . . . , ηn, 1, . . . , n) with ηj ∈ E, j 0 such that the intervals (ηj − j, ηj + j) are pairwise disjoint and 0 ∈ n j=1[ηj − j, ηj + j] ⊂ (−R, R). Given a set K ⊂ (−R, 0) ∪ (0, R) we say that (η1, . . . , ηn, 1, . . . , n) covers K, if K ⊂ n j=1(ηj − j, ηj + j). With the help of these notations the basic estimate can be formulated as follows. Lemma 3.2 (basic estimate). There exist functions an, bn : Jn → (0, ∞) satisfying for all n, p ∈ N
- 52. Countable Spectrum, Transfinite Induction and Stability 39 (a) an(η1, . . . , ηn, 1, . . . , n) → 1 as ( 1, . . . , n) → 0 in Rn (b) an+p(η1, . . . , ηn+p, 1, . . . , n+p) → an(η1, . . . , ηn, 1, . . . , n) as ( n+1, . . . , n+p) → 0 in Rp (c) bn(η1, . . . , ηn, 1, . . . , n) → 0 as ( 1, . . . , n) → 0 in Rn (d) bn+p(η1, . . . , ηn+p, 1, . . . , n+p) → bn(η1, . . . , ηn, 1, . . . , n) as ( n+1, . . . , n+p) → 0 in Rp such that the following holds: If E is covered by (η1, . . . , ηn, 1, . . . , n) ∈ Jn then lim sup t→∞ t 0 f(s) ds ≤ 2f∞ R an(η1, . . . , ηn, 1, . . . , n) + bn(η1, . . . , ηn, 1, . . . , n). These estimates are obtained by changing the contour in the proof of The- orem 2.4 on the straight line i[−R, R] by introducing semicircles of radius j, j = 1, . . . , n. For the proof we refer to [AP92, Lemma 5.2] (which is a modification of [AB88, Lemma 3.1]). Remark. The reader might better understand the proof of [AP92, Lemma 5.2] by replacing “and 0 =” on line 11, 12 of p. 430 by a “−” and lifting the term to the end of line 10. Also the signs “+” on lines 15 and 17 should be replaced by a “−”. Proof of Theorem 3.1. Let E0 := E ∩[−R, R]. Thus E0 is compact and countable. Given an ordinal α we define Eα inductively by Eα = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ the set of all cluster points of Eα, if α is a successor ordinal; βα Eβ, if α is a limit ordinal. We will prove that the following statement S(α) holds for all ordinals α: S(α) : if Eα = ∅, then lim sup t→∞ t 0 f(s) ds ≤ 2 R f∞ (3.2) and if Eα is covered by (η1, . . . , ηn, 1, . . . , n) ∈ Jn then lim sup t→∞ t 0 f(s) ds ≤ 2f∞ R an(η1, . . . , ηn, 1, . . . , n) + bn(η1, . . . , ηn, 1, . . . , n). (3.3) Once this statement is proved the proof of the theorem is completed as fol- lows: Since Eα is compact and countable it possesses an isolated point whenever Eα is non empty. Thus Eα+1 Eα whenever Eα = ∅. This implies that Eα0 = ∅ for some α0 (see Proposition 5.2). Hence statement S(α0) gives the result.
- 53. 40 W. Arendt Now we prove that S(α) holds for all ordinals α. α = 0: If E0 = ∅, this is Theorem 2.4. If E0 = ∅, then this follows immediately from the basic estimate Lemma 3.2. α 0: Assume that S(β) holds for all β α. We show that S(α) holds. First case: α is a limit ordinal. Then Eα = βα Eβ. If Eα = ∅, then there exists β α such that Eβ = ∅. The inductive hypothesis implies that (3.2) holds. If Eα is covered by (η1, . . . , ηn, 1, . . . , n) ∈ Jn, then there exists β α such that Eβ is covered by (η1, . . . , ηn, 1, . . . , n). Thus (3.3) follows by the inductive hypothesis. Second case: α is a successor ordinal. If Eα = ∅, then Eα−1 is finite, say Eα−1 = {η1, . . . , ηn}. Choose j 0 so small that (η1, . . . , ηn, 1, . . . , n) ∈ Jn. Then it follows by the inductive hypothesis that lim sup t→∞ t 0 f(s) ds ≤ 2f∞ R an(η1, . . . , ηn, 1, . . . , n) + bn(η1, . . . , ηn, 1, . . . , n). Letting ( 1, . . . , n) → 0 in Rn yields (3.2). If Eα is covered by (η1, . . . , ηn, 1, . . . , n) ∈ Jn, then Eα−1 n j=1 (ηj − j, ηj + j) is finite, consisting of, say, {ηn+1, . . . , ηn+p}. Choose j 0, j = n+1, . . ., n+p, so small that (η1, . . . , ηn+p, 1, . . . , n+p) ∈ Jn+p. Then by the inductive hypothesis lim sup t→∞ t 0 f(s) ds ≤ 2f∞ R an+p(η1, . . . , ηn+p, 1, . . . , n+p) + bn+p(η1, . . . , ηn+p, 1, . . . , n+p). Sending ( n+1, . . . , n+p) to 0 in Rp gives the desired estimate (3.3). Thus S(α) is proved. Remark 3.3. Applying Theorem 3.1 to the function f(·+s) instead of f one obtains the estimate lim sup t→∞ t 0 f(s) ds − F∞ ≤ 2 R lim sup t→∞ f(t) (3.4) which improves (3.1), cf. [AP92, Remark 3.2]. The following is an immediate consequence of Theorem 3.1.
- 54. Countable Spectrum, Transfinite Induction and Stability 41 Corollary 3.4. Let E ⊂ R be closed and countable such that 0 ∈ E. Let F∞ ∈ X. Assume that (a) ˆ f(λ)−F∞ λ has a continuous extension to C+ iE and that (b) supt≥0 t 0 e−iηs f(s) ds ∞ for all η ∈ E. Then limt→∞ t 0 f(s) ds = F∞. Remark. In the case where (a) is replaced by the stronger hypothesis (a ) ˆ f has a holomorphic extension to an open set containing C+ iE, Corollary 3.4 is proved by Batty, van Nerven and Räbiger [BvNR98, Theorem 4.3], where a slightly weaker hypothesis than (b) is considered (cf. [BvNR98, Remark 2]. The methods are very different though. We may transform Corollary 3.4 into a Tauberian theorem of a different type where convergence of f(t) as t → ∞ is the conclusion. Corollary 3.5. Let f ∈ L∞ (0, ∞; X) and f∞ ∈ X. Assume that ˆ f(λ) − f∞ λ (Re λ 0) has a continuous extension to C+ iE where E ⊂ R is closed, countable and 0 ∈ E. Assume that sup t≥0 t 0 e−iηs f(s) ds ∞ for all η ∈ E. (3.5) Then limt→∞ 1 δ δ+t t f(s) ds = f∞ for all δ 0. If f is uniformly continuous on [τ, ∞) for some τ 0, then lim t→∞ f(t) = f∞. This follows from Corollary 3.4 as [AP92, Theorem 3.5] follows from [AP92, Theorem 3.1]. We refer to Chill [Chi98], [ABHN11, Theorem 4.9.7] for a different approach via Fourier Analysis to such Tauberian theorems. Finally, we apply Corollary 3.5 to power series. By D := {z ∈ C : |z| 1} we denote the unit disc and by Γ := {z ∈ C : |z| = 1} the unit circle. Corollary 3.6. Let an ∈ X, supn∈N0 an ∞, p(z) = ∞ n=0 anzn for z ∈ D. Let F be a closed, countable subset of Γ such that (a) p has a continuous extension to D F and (b) supN∈N N n=0 anzn ∞ for all z ∈ Γ. Then limn→∞ an = 0 It follows from Riesz’ Theorem 2.5 that ∞ n=0 anzn = p(z) for all z ∈ D F.
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- 59. The Project Gutenberg eBook of Faux's Memorable Days in America, 1819-20; and Welby's Visit to North America, 1819-20, part 2 (1820)
- 60. This ebook is for the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this ebook or online at www.gutenberg.org. If you are not located in the United States, you will have to check the laws of the country where you are located before using this eBook. Title: Faux's Memorable Days in America, 1819-20; and Welby's Visit to North America, 1819-20, part 2 (1820) Author: W. Faux Adlard Welby Editor: Reuben Gold Thwaites Release date: March 18, 2013 [eBook #42364] Most recently updated: October 23, 2024 Language: English Credits: Produced by Greg Bergquist and the Online Distributed Proofreading Team at http://www.pgdp.net (This file was produced from images generously made available by The Internet Archive/American Libraries.) *** START OF THE PROJECT GUTENBERG EBOOK FAUX'S MEMORABLE DAYS IN AMERICA, 1819-20; AND WELBY'S VISIT TO NORTH AMERICA, 1819-20, PART 2 (1820) ***
- 61. Early Western Travels 1748-1846 Volume XII Early Western Travels 1748-1846 A Series of Annotated Reprints of some of the best and rarest contemporary volumes of travel, descriptive of the Aborigines and Social and Economic Conditions in the Middle and Far West, during the Period of Early American Settlement Edited with Notes, Introductions, Index, etc., by Reuben Gold Thwaites, LL.D. Editor of The Jesuit Relations and Allied Documents, Original Journals of the Lewis and Clark Expedition, Hennepin's New Discovery, etc. Volume XII Part II (1820) of Faux's Memorable Days in America, 1819-20; and Welby's Visit to North America, 1819-20.
- 62. Cleveland, Ohio The Arthur H. Clark Company 1905 Copyright 1905, by THE ARTHUR H. CLARK COMPANY ALL RIGHTS RESERVED The Lakeside Press R. R. DONNELLEY SONS COMPANY CHICAGO
- 63. CONTENTS OF VOLUME XII I Memorable Days in America: being a Journal of a Tour to the United States, etc. (Part II: January 1-July 21, 1820.) William Faux 11 II A Visit to North America and the English Settlements in Illinois, with a Winter Residence at Philadelphia; solely to ascertain the actual prospects of the Emigrating Agriculturist, Mechanic, and Commercial Speculator. Adlard Welby Author's Dedication 145 Author's Preface 147 Text The Voyage 151 Ship Cookery 156 Situation of a Passenger on board ship 156 Drive to the Falls of the Passaic River, Jersey State 166 Philadelphia 172 A Pensilvanian Innkeeper 188 American Waiters 196 Servants 199
- 64. Black Population in Free Pensilvania 200 Night 200 Americans and Scots 201 Virginia 202 Wheeling 204 State of Ohio 205 Kentucky. Maysville or Limestone 214 An Odd Mistake 218 Lexington, Kentucky 221 Frankfort 222 Louisville 226 Indiana 227 Vincennes (Indiana) 236 A Visit to the English Settlement in the Illinois 248 Harmony 260 A Winter at Philadelphia 294 Horrible Execution! 309 Lectures on Anatomy 314
- 65. ILLUSTRATIONS TO VOLUME XII Log Tavern, Indiana 142 Facsimile of title-page to Welby 143 Little Brandywine, Pennsylvania 176 Bridge at Columbia, Pennsylvania 179 Susquehannah River at Columbia 184 Place of Worship Burial Ground, at Ligonier Town, Pennsylvania 185 Widow McMurran's Tavern, Scrub Ridge 189 View on Scrub Ridge 193 Wooden scoop (text cut) 203 Ferry at Maysville, on the Ohio 209 Maysville, on the Ohio, Kentucky 215 Frankfort, Kentucky 224 The Church at Harmonie 264 Bridge at Zanesville, Ohio 277 View at Fort Cumberland, Maryland 281 View at Fort Cumberland, Maryland 286 Part II (1820) of Faux's Memorable Days in America November 27, 1818-July 21, 1820
- 66. Reprint of the original edition: London, 1823. Part I is comprised in Volume XI of our series
- 67. JOURNAL (PART II) January 1st, 1820.—I left Princeton at ten o'clock, with Mr. Phillips and Mr. Wheeler; and here parted with my good and kind friend Ingle. I met and spoke, ten miles off, with two hog-jobbing judges, Judge Prince and Judge Daniel,104 driving home twenty fat hogs, which they had just bought. I reached, and rested at Petersburgh,105 consisting of fifteen houses. I passed good farms. Our landlord of this infant town, though having an [333] ostler, was compelled to groom, saddle, unsaddle, and to do all himself. Having fifty dollars owing to him, from a gentleman of Evansville, he arrested him, when he went into the bounds; then he sued one of the bondsmen, who also entered the bounds. The squire is next to be sued, who, it is expected, will do likewise. Sunday, 2nd.—I rode thirty-one miles this day, and rested at Edmonstone, in a little cold log-hole, out of which I turned an officer's black cat, which jumped from the roof into our faces, while in bed; but she soon found her way in again, through a hole in the roof. The cat liked our fire. We got no coffee nor tea, but cold milk and pork, and corn cake. 3rd.—Travelled all day, through the mud-holes formed by springs running from countless hills, covered with fine timber, to breakfast, at three o'clock, p. m. I supped and slept at Judge Chambers's, a comfortable house, and saw again the judge's mother, of eighty, whose activity and superior horsemanship, I have before mentioned. I smoked a segar with Mrs. Judge, while she smoked her pipe, (the first pipe I have seen here.) She, as well as the old lady, is a quaker.
- 68. The judge was gone to the metropolitan town of Coridon, being a senator, on duty.106 The land which I passed over all this day, seemed poor, but full of wild turkeys and bears. 4th.—I reached Miller's to supper, but found no [334] coffee; cold milk only, as a substitute. The ride hither is interesting, through a fine rolling country. The wolves howled around us all night. 5th.—Passed the Silver Hills,107 from the summit of which is a fine, extensive prospect of Kentucky, the Ohio, and of Louisville, where we breakfasted. I called with Mr. Flower's letter to Archer, who was out. I received the present of a cow-hide whip, from a lady, and promised to treat the beast kindly, for her sake. Judge Waggoner recently shook hands at a whiskey-shop, with a man coming before him that day, to be tried for murder. He drank his health, and wished him well through. I rode seven miles with an intelligent old Kentucky planter, having four children, who cultivate his farm, without negroes. He says, Kentucky is morally and physically ruined. We have been brought up to live without labour: all are demoralized. No man's word or judgment is to be taken for the guidance and government of another. Deception is a trade, and all are rogues. The west has the scum of all the earth. Long ago it was said, when a man left other States, he is gone to hell, or Kentucky. The people are none the better for a free, good government. The oldest first settlers are all gone or ruined. Your colt, sir, of one hundred dollars, is worth only fifteen dollars. At Louisville, as good a horse can be bought at ten dollars, or fifteen dollars. You are therefore cheated. The Missouri territory boasts the best land in [335] the country, but is not watered by springs. Wells are, however, dug, abounding in good water, says our hearty landlord, just returned from viewing that country. The bottom land is the finest in the world. Corn, from sixty to eighty bushels, and wheat, from forty to sixty bushels an acre. The best prairies are full of fine grass, flowers, and weeds, not coarse, benty, sticky grass, which denotes the worst of prairie land. Grass, of
- 69. a short fine quality, fit for pasture or hay, every where abounds. The country is full of wild honey, some houses having made seven and eight barrels this season, taken out of the trees, which are cut down without killing the bees. These industrious insects do not sting, but are easily hived and made tame. Our landlord likes the Missouri, but not so well as Old Kentucky. Two grim, gaunt-looking men burst into our room, at two, this morning; and by six, the landlord disturbed us by cow-hiding his negro, threatening to squeeze the life out of him. 6th.—I rode all day through a country of fine plantations, and reached Frankfort to supper, with the legislative body, where I again met my gay fellow-traveller, Mr. Cowen. It was interesting to look down our table, and contemplate the many bright, intelligent faces around me: men who might honour any nation. As strangers, we were [336] invited by the landlord, (the best I have seen) to the first rush for a chance at the table's head. 7th.—I travelled this day through a fine country of rich pasture and tillage, to Lexington City, to Keen's excellent tavern. I drank wine with Mr. Lidiard, who is removing eastward, having spent 1,100l. in living, and travelling to and fro. Fine beef at three cents per lb. Fat fowls, one dollar per dozen. Who would not live in old Kentucky's first city? 8th.—Being a wet day, I rested all day and this night. Prairie flies bleed horses nearly to death. Smoke and fire is a refuge to these distressed animals. The Indian summer smoke reaches to the Isle of Madeira. Visited the Athenæum. Viewed some fine horses, at two hundred dollars each. Sunday, 9th.—I quitted Lexington, and one of the best taverns in America, for Paris, Kentucky, and a good, genteel farm-house, the General Washington, twenty-three miles from the city, belonging to Mr. Hit, who, though owning between four hundred and five hundred acres of the finest land in Kentucky, does not think it beneath him to entertain travellers and their horses, on the best fare and beds in
- 70. the country. He has been offered sixty dollars, and could now have forty dollars an acre, for his land, which averages thirty bushels of wheat, and sixty bushels of corn per acre, and, in [337] natural or artificial grass, is the first in the world. Sheep, (fine stores) one dollar per head; beef, fine, three cents per lb., and fowls, one dollar per dozen. 10th.—Rode all day in the rain and mud, and through the worst roads in the universe, frequently crossing creeks, belly deep of our horses. Passed the creek at Blue-lick, belly deep, with sulphurous water running from a sulphur spring, once a salt spring. The water stinks like the putrid stagnant water of an English horse-pond, full of animal dung. This is resorted to for health. Five or six dirkings and stabbings took place, this fall, in Kentucky. 11th.—Breakfasted at Washington, (Kentucky) where we parted with Mr. Phillips, and met the Squire, and another gentleman, debating about law. Rested at Maysville, a good house, having chambers, and good beds, with curtains. The steam-boats pass this handsome river town, at the rate of fifteen to twenty miles an hour. To the passenger, the effect is beautiful, every minute presenting new objects of attraction. 12th.—Crossed the Ohio in a flat, submitting to Kentuckyan imposition of seventy-five cents a horse, instead of twenty-five, because we were supposed to be Yankees. We will not, said the boat-man, take you over, for less than a dollar each. We heard of you, yesterday. The gentleman in the cap (meaning me) looks as though he [338] could afford to pay, and besides, he is so slick with his tongue. The Yankees are the smartest of fellows, except the Kentuckyans. Sauciness and impudence are characteristic of these boat-men, who wished I would commence a bridge over the river. Reached Union town, Ohio,108 and rested for the night. 13th.—Breakfasted at Colonel Wood's. A fine breakfast on beef, pork-steaks, eggs, and coffee, and plenty for our horses, all for fifty cents each. Slept at Colonel Peril's, an old Virginian revolutionary soldier, living on 400 acres of fine land, in a good house, on an
- 71. eminence, which he has held two years only. He now wishes to sell all at ten dollars an acre, less than it cost him, because he has a family who will all want as much land each, in the Missouri, at two dollars. He never had a negro. He knows us to be English from our dialect. We passed, this day, through two or three young villages. 14th.—Breakfasted at Bainbridge,109 where is good bottom land, at twenty to thirty dollars an acre, with improvements. The old Virginian complains of want of labourers. A farmer must do all himself. Received of our landlady a lump of Ohio wild sugar, of which some families make from six to ten barrels a-year, sweet and good enough. Reached Chilicothé, on the Sciota river, to [339] sup and rest at the tavern of Mr. Madera, a sensible young man. Here I met Mr. Randolph, a gentleman of Philadelphia, from Missouri and Illinois, who thinks both sickly, and not to be preferred to the east, or others parts of the west. I saw three or four good houses, in the best street, abandoned, and the windows and doors rotting out for want of occupants. 15th.—I rode all day through a fine interesting country, abounding with every good thing, and full of springs and streams. Near Lancaster,110 I passed a large high ridge of rocks, which nature has clothed in everlasting green, being beautified with the spruce, waving like feathers, on their bleak, barren tops. I reached Lancaster to rest; a handsome county seat, near which land is selling occasionally from sixteen dollars to twenty dollars. A fine farm of 170 acres, 100 being cleared, with all improvements, was sold lately by the sheriff, at sixteen dollars one cent an acre, much less than it cost. Labour is to be had at fifty cents and board, but as the produce is so low, it is thought farming, by hired hands, does not pay. Wheat, fifty cents; corn, 33½ cents; potatoes, 33 cents a bushel; beef, four dollars per cwt.; pork, three dollars; mutton none; sheep being kept only for the wool, and bought in common at 2s. 8d. per head. Met Judge and General ——, who states that four millions of acres of land will this year [340] be offered to sale, bordering on the lakes.
- 72. Why then should people go to the Missouri? It is not healthy near the lakes, on account of stagnant waters, made by sand bars, at the mouth of lake rivers. The regular periodical rising and falling of the lakes is not yet accounted for. There is no sensible diminution, or increase of the lake-waters. A grand canal is to be completed in five years, when boats will travel.111 Sunday, 16th.—I left Lancaster at peep of day, travelling through intense cold and icy roads to Somerset, eighteen miles, in five hours, to breakfast.112 Warmed at an old quarter-section man, a Dutch American, from Pennsylvania. He came here eleven years since, cleared seventy acres, has eight children, likes his land, but says, produce is too low to make it worth raising. People comfortably settled in the east, on good farms, should stay, unless their children can come and work on the land. He and his young family do all the work. Has a fine stove below, warming the first, and all other floors, by a pipe passing through them. I slept at a good tavern, the keeper of which is a farmer. All are farmers, and all the best farmers are tavern-keepers. Farms, therefore, on the road, sell from 50 to 100 per cent more than land lying back, though it is no better in quality, and for mere farming, worth no more. But on the road, a farm and frequented tavern is found to be [341] a very beneficial mode of using land; the produce selling for double and treble what it will bring at market, and also fetching ready money. Labour is not to be commanded, says our landlord. 17th.—Started at peep of day in a snow-storm, which had covered the ground six inches deep. Breakfasted at beautiful Zanesville, a town most delightfully situated amongst the hills. Twelve miles from this town, one Chandler, in boring for salt, hit upon silver; a mine, seven feet thick, 150 feet below the surface. It is very pure ore, and the proprietor has given up two acres of the land to persons who have applied to the legislature to be incorporated. He is to receive one-fifth of the net profits.
- 73. 18th.—I rode all day through a fine hilly country, full of springs and fountains. The land is more adapted for good pasture than for cultivation. Our landlord, Mr. Gill, states that wheat at fifty cents is too low; but, even at that price, there is no market, nor at any other. In some former years, Orleans was a market, but now it gets supplied from countries more conveniently situated than Ohio, from which it costs one dollar, or one dollar and a quarter per barrel, to send it. Boats carrying from 100 to 500 barrels, sell for only 16 dollars. From a conversation, with an intelligent High Sheriff of this county, I learn that no common debtor has ever lain in prison longer than five [342] days. None need be longer in giving security for the surrender of all property. 19th.—Reached Wheeling late at night, passing through a romantic, broken, mountainous country, with many fine springs and creeks. Thus I left Ohio, which, thirty years ago, was a frontier state, full of Indians, without a white man's house, between Wheeling, Kaskasky, and St. Louis. 20th.—Reached Washington, Pennsylvania, to sleep, and found our tavern full of thirsty classics, from the seminary in this town. 21st.—Reached Pittsburgh, through a beautiful country of hills, fit only for pasture. I viewed the fine covered bridges over the two rivers Monongahela and Allegany, which cost 10,000 dollars each. The hills around the city shut it in, and make the descent into it frightfully precipitous. It is most eligibly situated amidst rocks, or rather hills, of coal, stone, and iron, the coals lying up to the surface, ready for use. One of these hills, or coal banks, has been long on fire, and resembles a volcano. Bountiful nature has done every thing for this rising Birmingham of America. We slept at Wheeling, at the good hotel of Major Spriggs, one of General Washington's revolutionary officers, now near 80, a chronicle of years departed.113 22nd.—Bought a fine buffalo robe for five dollars. [343] The buffaloes, when Kentucky was first settled, were shot, by the
- 74. settlers, merely for their tongues; the carcase and skin being thought worth nothing, were left where the animal fell. Left Pittsburgh for Greensburgh, travelling through a fine, cultivated, thickly settled country, full of neat, flourishing, and good farms, the occupants of which are said to be rich. Land, on the road, is worth from fifteen to thirty dollars; from it, five to fifteen dollars per acre. The hills and mountains seem full of coal-mines and stone- quarries, or rather banks of coal and stone ever open gratuitously to all. The people about here are economical and intelligent; qualities characteristic of Pennsylvania. Sunday, 23d.—We agreed to rest here until the morrow; finding one of our best horses sick; and went to Pittsburgh church. 24th.—My fellow traveller finding his horse getting worse, gave him away for our tavern bill of two days, thus paying 175 dollars for two days board. While this fine animal remained ours, no doctor could be found, but as soon as he became our landlord's, one was discovered, who engaged to cure him in a week. Mr. Wheeler took my horse, and left me to come on in the stage, to meet again at Chambersburgh. The country round about here is fine, but there is no market, except at Baltimore, at five dollars a barrel for flour. The carriage costs two and half [344] dollars. I saw two young ladies, Dutch farmers' daughters, smoking segars in our tavern, very freely, and made one of their party. Paid twelve dollars for fare to Chambersburgh.114 Invited to a sleying party of ten gentlemen, one of whom was the venerable speaker (Brady) of the senate of this state. They were nearly all drunk with apple-toddy, a large bowl of which was handed to every drinker. One gentleman returned with a cracked skull. 25th.—Left this town, at three o'clock in the morning, in the stage, and met again at Bedford, and parted, perhaps, for ever, with my agreeable fellow-traveller, Mr. Wheeler, who passed on to New York. Passed the Laurel-hill, a huge mountain, covered with everlasting
- 75. green, and a refuge for bears, one of which was recently killed with a pig of 150lbs. weight in his mouth. 26th.—Again mounted my horse, passing the lonely Allegany mountains, all day, in a blinding snow-storm, rendering the air as dense as a November fog in London. Previous to its coming on, I found my naked nose in danger. The noses of others were wrapped up in flannel bags, or cots, and masks for the eyes, which are liable to freeze into balls of ice. Passed several flourishing villages. The people here seem more economical and simple, than in other states. Rested at M'Connell's town, 100 miles from Washington city. [345] 27th.—Crossed the last of the huge Allegany mountains, called the North Mount, nine miles over, and very high. My horse was belly deep in snow. Breakfasted at Mercersburgh, at the foot of the above mountain, and at the commencement of that fine and richest valley in the eastern states, in which Hagar's town stands, and which extends through Pennsylvania, Maryland, and Virginia, from 100 to 200 miles long, and from 30 to 40 broad. Land here, three years ago, sold at 100 to 120 dollars, although now at a forced sale, 160 acres sold for only 1,600 dollars, with improvements, in Pennsylvania. And if, says my informant, the state makes no law to prevent it, much must come into the market, without money to buy, except at a ruinous depreciation. Passed Hagar's town, to Boonsburgh, to rest all night, after 37 miles travel. The old Pennsylvanian farmer, in answer to How do you do without negroes? said, Better than with them. I occupy of my father 80 acres in this valley, and hire all my hands, and sell five loads of flour, while some of the Marylanders and Virginians cannot raise enough to maintain their negroes, who do but little work. 28th.—Breakfasted on the road; passed Middletown, with two fine spires, a good town; and also Frederick town, a noble inland town, and next to Lancaster, in Pennsylvania, and the first [346] in the
- 76. United States. It has three beautiful spires. It is much like a second rate English town, but not so cleanly; something is dirty, or in ruins. It stands at the foot of the Blue Ridge, in the finest, largest vale in the world, running from the eastern sea to the Gulf of Mexico. Rested at Windmiller's, a stage-house, thirty miles from Washington, distinguished only by infamous, ungenerous, extortion from travellers. Here I paid 75 cents for tea; 25 cents for a pint of beer, 9s. sterling for a bushel of oats and corn, and 50 cents for hay for the night. The horse cost 6s. 9d. in one night. 29th.—Rode from seven till eleven o'clock, sixteen miles to breakfast, at Montgomery-court-house, all drenched in rain. I reached Washington city, at six this evening. Here, for the first time, I met friend Joseph Lancaster, full of visionary schemes, which are unlikely to produce him bread. Sunday, 30th.—Went to Congress-hall, and heard grave senators wrangling about slavery. Governor Barbour spoke with eloquence. Friend Lancaster's daily and familiar calls on the great, and on his Excellency, the President, about schooling the Indians, and his praises of the members, are likely to wear out all his former fame, already much in ruins. I was this day introduced by him to —— Parr, Esq., an English gentleman of fortune, from Boston, Lincolnshire, [347] who has just returned from a pedestrian pilgrimage to Birkbeck and the western country. February 1st.—I again went to Congress, where I heard Mr. Randolph's good speech on the Missouri question. This sensible orator continually refers to English authors and orators, insomuch that all seemed English. These American statesmen cannot open their mouths without acknowledging their British origin and obligations.—I shall here insert some observations on the constitution and laws of this country, and on several of the most distinguished members of Congress, for which I am indebted to the pen of G. Waterstone, Esq., Congressional Librarian at Washington.115
- 77. Observations on the Constitution and Laws of the United States, with Sketches of some of the most prominent public Characters. Like the Minerva of the ancients, the American people have sprung, at once, into full and vigorous maturity, without the imbecility of infancy, or the tedious process of gradual progression. They possess none of the thoughtless liberality and inconsiderate confidence of youth; but are, already, distinguished by the cold and cautious policy of declining life, rendered suspicious by a long acquaintance with the deceptions and the vices of the world. Practitioners of jurisprudence have become [348] almost innumerable, and the great end of all laws, the security and protection of the citizen, is in some degree defeated. It is to the multiplicity and ambiguity of the laws of his age, that Tacitus has ascribed most of the miseries which were then experienced; and this evil will always be felt where they are ambiguous and too numerous. In vain do the Americans urge that their laws have been founded on those of England, the wisdom and excellence of which have been so highly and extravagantly eulogized. The difference, as Mably correctly observes, between the situation of this country and that is prodigious;116 the government of one having been formed in an age of refinement and civilization, and that of the other, amidst the darkness and barbarism of feudal ignorance. In most of the states the civil and criminal code is defective; and the latter, like that of Draco, is often written in blood. Why should not each state form a code of laws for itself, and cast off this slavish dependence on Great Britain, whom they pretend so much to dislike? With a view of explaining more perfectly the nature of this constitution, I will briefly exhibit the points in which the British and American governments differ [349]. In England. In America. I. The king possesses imperial dignity. There is no king; the president acts as
- 78. the chief magistrate of the nation only. II. This imperial dignity is hereditary and perpetual. The presidency lasts only four years. III. The king has the sole power of making war and peace, and of forming treaties with foreign powers. The president can do neither, without the consent of Congress. IV. The king alone can levy troops, build fortresses, and equip fleets. The president has no such power: this is vested in Congress. V. He is the source of all judicial power, and the head of all the tribunals of the nation. The executive has only the appointment of judges, with the consent of the senate, and is not connected with the judiciary. VI. He is the fountain of all honour, office, and privilege; can create peers, and distribute titles and dignities. The president has no such power. There are no titles, and he can only appoint to office, by and with the consent of the senate. VII. He is at the head of the national church, and has supreme control over it. There is no established church. VIII. He is the superintendent of commerce; regulates The president has no such power.
- 79. the weights and measures, and can alone coin money and give currency to foreign coin. IX. He is the universal proprietor of the kingdom. The president has nothing to do with the property of the United States [350]. X. The king's person is sacred and inviolate; he is accountable to no human power, and can do no wrong. The president is nothing more than an individual, is amenable like all civil officers, and considered as capable of doing wrong as any other citizen. XI. The British legislature contains a house of lords, 300 nobles, whose seats, honours, and privileges are hereditary. There are no nobles, and both houses of Congress are elected. It may, perhaps, be unnecessary to adduce more points of difference to illustrate the nature of the American government. These are amply sufficient to demonstrate the entire democratic tendency of the constitution of the United States, and the error under which those persons labour, who believe that but few differences, and those immaterial and unimportant, exist between these two governments. They have, indeed, in common the Habeas Corpus and the Trial by Jury, the great bulwarks of civil liberty, but in almost every other particular they disagree. The second branch of this government is the legislature. This consists of a Senate and House of Representatives; the members of
- 80. the latter are chosen every two years by the people; and those of the former, every six years by the legislatures of the different states. It is in this branch that the American government differs from the republics of ancient and modern times; it is this which [351] makes it not a pure, but a representative democracy; and it is this which gives it such a decided superiority over all the governments in the world. Experience has demonstrated the impracticability of assembling a numerous collection of people to frame laws, and their incompetency, when assembled, for judicious deliberation and prompt and unbiassed decision. The passions of illiterate and unthinking men are easily roused into action and inflamed to madness. Artful and designing demagogues are too apt to take advantage of those imbecilities of our nature, and to convert them to the basest purposes. The qualifications of representatives are very simple. It is only required that they should be citizens of the United States, and have attained the age of twenty-five. The moment their period of service expires, they are again, unless re-elected, reduced to the rank and condition of citizens. If they should have acted in opposition to the wishes and interests of their constituents, while performing the functions of legislation, the people possess the remedy and can exercise it without endangering the peace and harmony of society; the offending member is dropped, and his place supplied by another, more worthy of confidence. This consciousness of responsibility, on the part of the representatives, operates as a perpetual guarantee to the people, and protects and secures them in the enjoyment of their political and civil liberties. [352] It must be admitted that the Americans have attained the Ultima Thulé in representative legislation, and that they enjoy this inestimable blessing to a much greater extent than the people of Great Britain. Of the three distinct and independent branches of that government, one only owes its existence to the free suffrages of the people, and this, from the inequality of representation, the long intervals between the periods of election, and the liability of members, from this circumstance, to be corrupted, is not so
- 81. important and useful a branch as might otherwise be expected. Imperfect, however, as it is, the people, without it, would indeed be slaves, and the government nothing more than a pure monarchy. The American walks abroad in the majesty of freedom; if he be innocent, he shrinks not from the gaze of upstart and insignificant wealth, nor sinks beneath the oppression of his fellow-man. Conscious of his rights and of the security he enjoys, by the liberal institutions of his country, independence beams in his eye, and humanity glows in his heart. Has he done wrong? He knows the limits of his punishment, and the character of his judges. Is he innocent? He feels that no power on earth can crush him. What a condition is this, compared with that of the subjects of almost all the European nations! As long as it is preserved, the security of the citizen and the union of the states, will be guaranteed, [353] and the country thus governed, will become the home of the free, the retreat of misery, and the asylum of persecuted humanity. As a written compact, it is a phenomenon in politics, an unprecedented and perfect example of representative democracy, to which the attention of mankind is now enthusiastically directed. Most happily and exquisitely organized, the American constitution is, in truth, at once a monument of genius, and an edifice of strength and majesty. The union of its parts forms its solidity, and the harmony of its proportions constitutes its beauty. May it always be preserved inviolate by the gallant and highminded people of America, and may they never forget that its destruction will be the inevitable death-blow of liberty, and the probable passport to universal despotism! The speaker of the House of Representatives is Mr. Clay, a delegate from Kentucky, and who, not long ago, acted a conspicuous part, as one of the American commissioners at Ghent.117 He is a tall, thin, and not very muscular man; his gait is stately, but swinging; and his countenance, while it indicates genius, denotes dissipation. As an orator, Mr. Clay stands high in the estimation of his countrymen, but he does not possess much gracefulness or elegance of manner; his eloquence is impetuous and vehement; it rolls like a
- 82. torrent, but like a torrent which is sometimes irregular, and occasionally obstructed. Though there is a [354] want of rapidity and fluency in his elocution, yet he has a great deal of fire and vigour in his expression. When he speaks he is full of animation and earnestness; his face brightens, his eye beams with additional lustre, and his whole figure indicates that he is entirely occupied with the subject on which his eloquence is employed. In action, on which Demosthenes laid such peculiar emphasis, and which was so highly esteemed among the ancients, Mr. Clay is neither very graceful nor very imposing. He does not, in the language of Shakespear, so suit the word to the action, and the action to the word, as not to o'erstep the modesty of nature. In his gesticulation and attitudes, there is sometimes an uniformity and awkwardness that lessen his merit as an orator, and in some measure destroy the impression and effect his eloquence would otherwise produce. Mr. Clay does not seem to have studied rhetoric as a science, or to have paid much attention to those artificial divisions and rhetorical graces and ornaments on which the orators of antiquity so strongly insist. Indeed, oratory as an art is but little studied in this country. Public speakers here trust almost entirely to the efficacy of their own native powers for success in the different fields of eloquence, and search not for the extrinsic embellishments and facilities of art. It is but rarely they unite the Attic and Rhodian manner, and still more rarely do they devote their attention to the acquisition [355] of those accomplishments which were, in the refined ages of Greece and Rome, considered so essential to the completion of an orator. Mr. Clay, however, is an eloquent speaker; and notwithstanding the defects I have mentioned, very seldom fails to please and convince. His mind is so organized that he overcomes the difficulties of abstruse and complicated subjects, apparently without the toil of investigation or the labour of profound research. It is rich, and active, and rapid, grasping at one glance, connections the most distant, and consequences the most remote, and breaking down the trammels of error and the cobwebs of sophistry. When he rises to speak he always commands attention, and almost always satisfies the mind on which his eloquence is intended to operate. The warmth and fervor
- 83. of his feelings, and the natural impetuosity of his character, which seem to be common to the Kentuckians, often indeed lead him to the adoption of opinions, which are not, at all times, consistent with the dictates of sound policy. Though ambitious and persevering, his intentions are good and his heart is pure; he is propelled by a love of country, but yet is solicitous of distinction; he wishes to attain the pinnacle of greatness without infringing the liberties, or marring the prosperity of that land of which it seems to be his glory to be a native. [356] The prominent traits of Mr. Clay's mind are quickness, penetration, and acuteness; a fertile invention, discriminating judgment, and good memory. His attention does not seem to have been much devoted to literary or scientific pursuits, unconnected with his profession; but fertile in resources, and abounding in expedients, he is seldom at a loss, and if he is not at all times able to amplify and embellish, he but rarely fails to do justice to the subject which has called forth his eloquence. On the most complicated questions, his observations made immediately and on the spur of the occasion, are generally such as would be suggested by long and deep reflection. In short, Mr. Clay has been gifted by nature with great intellectual superiority, which will always give him a decided influence in whatever sphere it may be his destiny to revolve. Mr. Clay's manners are plain and easy. He has nothing in him of that reserve which checks confidence, and which some politicians assume; his views of mankind are enlarged and liberal; and his conduct as a politician and a statesman has been marked with the same enlarged and liberal policy. As Speaker of the House of Representatives, he presides generally with great dignity, and decides on questions of order, sometimes, indeed, with too much precipitation, but almost always correctly. It is but seldom his decisions are disputed, [357] and when they are, they are not often reversed.118 A Statesman, says Mirabeau, presents to the mind the idea of a vast genius improved by experience, capable of embracing the mass
- 84. of social interests, and of perceiving how to maintain true harmony among the individuals of which society is composed, and an extent of information which may give substance and union to the different operations of government. Mr. Pinkney119 is between fifty and sixty years of age; his form is sufficiently elevated and compact to be graceful, and his countenance, though marked by the lines of dissipation, and rather too heavy, is not unprepossessing or repulsive. His eye is rapid in its motion, and beams with the animation of genius; but his lips are too thick, and his cheeks too fleshy and loose for beauty; there is too a degree of foppery, and sometimes of splendor, manifested in the decoration of his person, which is not perfectly reconcileable to our [358] ideas of mental superiority, and an appearance of voluptuousness about him which cannot surely be a source of pride or of gratification to one whose mind is so capacious and elegant. It is not improbable, however, that this character is assumed merely for the purpose of exciting a higher admiration of his powers, by inducing a belief that, without the labour of study or the toil of investigation, he can attain the object of his wishes and become eminent, without deigning to resort to that painful drudgery by which meaner minds and inferior intellects are enabled to arrive at excellence and distinction. At the first glance, you would imagine Mr. Pinkney was one of those butterflies of fashion, a dandy, known by their extravagant eccentricities of dress, and peculiarities of manners; and no one could believe, from his external appearance, that he was, in the least degree, intellectually superior to his fellow men. But Mr. Pinkney is indeed a wonderful man, and one of those beings whom the lover of human nature feels a delight in contemplating. His mind is of the very first order; quick, expanded, fervid, and powerful. The hearer is at a loss which most to admire, the vigour of his judgment, the fertility of his invention, the strength of his memory, or the power of his imagination. Each of these faculties he possesses in an equal degree of perfection, and each is displayed in its full maturity, when the [359] magnitude of the subject on which he descants renders its operation necessary. This
- 85. singular union of the rare and precious gifts of nature, has received all the strength which education could afford, and all the polish and splendour which art could bestow. Under the cloak of dissipation and voluptuousness his application has been indefatigable, and his studies unintermitted: the oil of the midnight lamp has been exhausted, and the labyrinths of knowledge have been explored. Mr. Pinkney is never unprepared, and never off his guard. He encounters his subject with a mind rich in all the gifts of nature, and fraught with all the resources of art and study. He enters the list with his antagonist, armed, like the ancient cavalier, cap-a-pee; and is alike prepared to wield the lance, or to handle the sword, as occasion may require. In cases which embrace all the complications and intricacies of law, where reason seems to be lost in the chaos of technical perplexity, and obscurity and darkness assume the dignified character of science, he displays an extent of research, a range of investigation, a lucidness of reasoning, and a fervor and brilliancy of thought, that excite our wonder, and elicit our admiration. On the driest, most abstract, and uninteresting questions of law, when no mind can anticipate such an occurrence, he occasionally blazes forth in all the enchanting exuberance of a chastened, but rich [360] and vivid imagination, and paints in a manner as classical as it is splendid, and as polished as it is brilliant. In the higher grades of eloquence, where the passions and feelings of our nature are roused to action, or lulled to tranquillity, Mr. Pinkney is still the great magician, whose power is resistless, and whose touch is fascination. His eloquence becomes sublime and impassioned, majestic and overwhelming. In calmer moments, when these passions are hushed, and more tempered feelings have assumed the place of agitation and disorder, he weaves around you the fairy circles of fancy, and calls up the golden palaces and magnificent scenes of enchantment. You listen with rapture as he rolls along: his defects vanish, and you are not conscious of any thing but what he pleases to infuse. From his tongue, like that of Nestor, language more sweet than honey flows; and the attention is constantly rivetted by the successive operation of the different
- 86. faculties of the mind. There are no awkward pauses, no hesitation for want of words or of arguments: he moves forward with a pace sometimes majestic, sometimes graceful, but always captivating and elegant. His order is lucid, his reasoning logical, his diction select, magnificent, and appropriate, and his style, flowing, oratorical, and beautiful. The most laboured and finished composition could not be better than that which he seems to utter [361] spontaneously and without effort. His judgment, invention, memory, and imagination, all conspire to furnish him at once with whatever he may require to enforce, embellish, or illustrate his subject. On the dullest topic he is never dry: and no one leaves him without feeling an admiration of his powers, that borders on enthusiasm. His satire is keen, but delicate, and his wit is scintillating and brilliant. His treasure is exhaustless, possessing the most extensive and varied information. He never feels at a loss; and he ornaments and illustrates every subject he touches. Nihil quod tetigit, non ornavit. He is never the same; he uses no common place artifice to excite a momentary thrill of admiration. He is not obliged to patch up and embellish a few ordinary thoughts, or set off a few meagre and uninteresting facts. His resources seem to be as unlimited as those of nature; and fresh powers, and new beauties are exhibited, whenever his eloquence is employed. A singular copiousness and felicity of thought and expression, united to a magnificence of amplification, and a purity and chastity of ornament, give to his eloquence a sort of enchantment which it is difficult to describe. Mr. Pinkney's mind is in a high degree poetical; it sometimes wantons in the luxuriance of its own creations; but these creations never violate the purity of classical taste and elegance. He [362] loves to paint when there is no occasion to reason; and addresses the imagination and passions, when the judgment has been satisfied and enlightened. I speak of Mr. Pinkney at present as a forensic orator. His career was too short to afford an opportunity of judging of his parliamentary eloquence; and, perhaps, like Curran, he might have failed in a field in which it was anticipated he would excel, or, at least, retain his usual pre-eminence. Mr. Pinkney, I think, bears a
- 87. stronger resemblance to Burke than to Pitt; but, in some particulars, he unites the excellences of both. He has the fancy and erudition of the former, and the point, rapidity, and elocution of the latter. Compared with his countrymen, he wants the vigour and striking majesty of Clay, the originality and ingenuity of Calhoun; but, as a rhetorician, he surpasses both. In his action, Mr. Pinkney has, unfortunately, acquired a manner, borrowed, no doubt, from some illustrious model, which is eminently uncouth and inelegant. It consists in raising one leg on a bench or chair before him, and in thrusting his right arm in a horizontal line from his side to its full length in front. This action is uniform, and never varies or changes in the most tranquil flow of sentiment, or the grandest burst of impassioned eloquence. His voice, though not naturally good, has been disciplined to modulation by art; and, if it is not always musical, it is [363] never very harsh or offensive. Such is Mr. Pinkney as an orator; as a diplomatist but little can be said that will add to his reputation. In his official notes there is too much flippancy, and too great diffuseness, for beauty or elegance of composition. It is but seldom that the orator possesses the requisites of the writer; and the fame which is acquired by the tongue sometimes evaporates through the pen. As a writer he is inferior to the present Attorney- General,120 who unites the powers of both in a high degree, and thus in his own person illustrates the position which he has laid down, as to the universality of genius. Mr. R. King is a senator from the State of New York, and was formerly the resident minister at the court of St. James's.121 He is now about sixty years of age, above the middle size, and somewhat inclined to corpulency. His countenance, when serious and thoughtful, possesses a great deal of austerity and rigour; but at other moments it is marked with placidity and benevolence. Among his friends he is facetious and easy; but when with strangers, reserved and distant; apparently indisposed to conversation, and inclined to taciturnity; but when called out, his colloquial powers are of no ordinary character, and his conversation becomes peculiarly instructive, fascinating, and humourous. Mr. King has read and
- 88. reflected much; and though long in public life, his attention [364] has not been exclusively devoted to the political sciences; for his information on other subjects is equally matured and extensive. His resources are numerous and multiplied, and can easily be called into operation. In his parliamentary addresses he always displays a deep and intimate knowledge of the subject under discussion, and never fails to edify and instruct if he ceases to delight. He has read history to become a statesman, and not for the mere gratification it affords. He applies the experience of ages, which the historical muse exhibits, to the general purposes of government, and thus reduces to practice the mass of knowledge with which his mind is fraught and embellished. As a legislator he is, perhaps, inferior to no man in this country. The faculty of close and accurate observation by which he is distinguished has enabled him to remark and treasure up every fact of political importance, that has occurred since the organization of the American government; and the citizen, as well as the stranger, is often surprised at the minuteness of his historical details, and the facility with which they are applied. With the various subjects immediately connected with politics, he has made himself well acquainted; and such is the strength of his memory, and the extent of his information, that the accuracy of his statements is never disputed. Mr. King, however, is somewhat of an [365] enthusiast, and his feelings sometimes propel him to do that which his judgment cannot sanction. When parties existed in this country, he belonged to, and was considered to be the leader of what was denominated the federal phalanx; and he has often, perhaps, been induced, from the influence of party feeling, and the violence of party animosity, to countenance measures that must have wounded his moral sensibilities; and that now, when reason is suffered to dictate, cannot but be deeply regretted. From a rapid survey of his political and parliamentary career, it would appear that the fury of party has betrayed him into the expression of sentiments, and the support of measures, that were, in their character, revolting to his feelings; but whatever he may have been charged with, his intentions, at least, were pure, and his exertions, as he conceived, calculated for the public good. He was indeed cried down by a class of emigrants from
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