![16 A. Böttcher, S.M. Grudsky and E.A. Maksimenko
be an orthonormal basis of eigenvectors such that Tn(a)v
(n)
j = λ
(n)
j v
(n)
j . The
present paper is dedicated to the asymptotic behavior of the eigenvectors v
(n)
j
as n → ∞.
To get an idea of the kind of results we will establish, consider the function
a(eix
) = 2−2 cosx. The range a(T) is the segment [0, 4]. It is well known that the
eigenvalues and eigenvectors of Tn(a) are given by
λ
(n)
j = 2 − 2 cos
πj
n + 1
, x
(n)
j =
2
n + 1
sin
mπj
n + 1
n
m=1
. (1.1)
(We denote the eigenvectors in this reference case by x
(n)
j and reserve the notation
v
(n)
j for the general case.) Let ϕ be the function
ϕ : [0, 4] → [0, π], ϕ(λ) = arccos
2 − λ
2
.
We have ϕ(λ
(n)
j ) = πj/(n + 1) and hence, apart from the normalization factor
2/(n + 1), x
(n)
j,m is the value of sin(mϕ(λ)) at λ = λ
(n)
j . In other words, an
eigenvector for λ is given by (sin(mϕ(λ)))n
m=1. A speculative question is whether
in the general case we can also find functions Ωm such that, at least asymptotically,
(Ωm(λ))n
m=1 is an eigenvector for λ. It turns out that this is in general impossible
but that after a slight modification the answer to the question is in the affirmative.
Namely, we will prove that, under certain assumptions, there are functions Ωm,
Φm and real-valued functions σ, η such that an eigenvector for λ = λ
(n)
j is always
of the form
Ωm(λ)+Φm(λ)+(−1)j+1
e−i(n+1)σ(λ)
e−iη(λ)
Φn+1−m(λ)+error term
n
m=1
. (1.2)
The error term will be shown to decrease to zero exponentially fast and uniformly
in j and m as n → ∞. Moreover, we will show that Ωm(λ) is an oscillating function
of m for each fixed λ and that Φm(λ) decays exponentially fast to zero as m → ∞
for each λ (which means that Φn+1−m(λ) is an exponentially increasing function
of m for each λ). Finally, it will turn out that
n
m=1
|Φm(λ)|2
n
m=1
|Ωm(λ)|2
= O
1
n
as n → ∞, uniformly in λ. Thus, the dominant term in (1.2) is Ωm(λ), while the
terms containing Φm(λ) and Φn+1−m(λ) may be viewed as twin babies.
If a is also an even function, a(eix
) = a(e−ix
) for all x, then all the matrices
Tn(a) are real and symmetric. In [4], we conjectured that then, again under addi-
tional but reasonable assumptions, the appropriately rotated extreme eigenvectors
v
(n)
j are all close to the vectors x
(n)
j . To be more precise, we conjectured that if
n → ∞ and j (or n − j) remains fixed, then there are complex numbers τ
(n)
j of](https://image.slidesharecdn.com/1021278-250521173013-9c3fc6a2/85/Recent-Trends-In-Toeplitz-And-Pseudodifferential-Operators-Duduchava-R-27-320.jpg)
![Eigenvectors of Large Hermitian Toeplitz Band Matrices 17
modulus 1 such that
τ
(n)
j v
(n)
j − x
(n)
j
2
= o(1), (1.3)
where · 2 is the 2
norm. Several results related to this conjecture were estab-
lished in [3] and [4]. We here prove this conjecture under assumptions that will be
specified in the following paragraph. We will even be able to show that the o(1)
in (1.3) is O(j/n) if j/n → 0 and O(1 − j/n) if j/n → 1.
Throughout what follows we assume that a is a Laurent polynomial
a(t) =
r
k=−r
aktk
(t = eix
∈ T)
with r ≥ 2, ar = 0, and ak = a−k for all k. The last condition means that a is
real-valued on T. We assume without loss of generality that a(T) = [0, M] with
M 0 and that a(1) = 0 and a(eiϕ0
) = M for some ϕ0 ∈ (0, 2π). We require that
the function g(x) := a(eix
) is strictly increasing on (0, ϕ0) and strictly decreasing
on (ϕ0, 2π) and that the second derivatives of g at x = 0 and x = ϕ0 are nonzero.
Finally, we denote by [α, β] ⊂ [0, M] a segment such that if λ ∈ [α, β], then the
2r − 2 zeros of the Laurent polynomial a(z) − λ that lie in C T are pairwise
distinct.
Note that we exclude the case r = 1, because in this case the eigenvalues
and eigenvectors of Tn(a) are explicitly available. Also notice that if r = 2, which
is the case of pentadiagonal matrices, then for every λ ∈ [0, M] the polynomial
a(z) − λ has two zeros on T, one zero outside T, and one zero inside T. Thus,
in this situation the last requirement of the previous paragraph is automatically
satisfied for [α, β] = [0, M].
The asymptotic behavior of the extreme eigenvalues and eigenvectors of
Tn(a), that is, of λ
(n)
j and v
(n)
j when j or n − j remain fixed, has been stud-
ied by several authors. As for extreme eigenvalues, the pioneering works are [7],
[9], [11], [12], [18], while recent papers on the subject include [3], [6], [8], [10], [13],
[14], [15], [19], [20]. See also the books [1] and [5]. Much less is known about the
asymptotics of the eigenvectors. Part of the results of [4] and [19] may be inter-
preted as results on the behavior of the eigenvectors “in the mean” on the one hand
and as insights into what happens if eigenvectors are replaced by pseudomodes on
the other. In [3], we investigated the asymptotics of the extreme eigenvectors of
certain Hermitian (and not necessarily banded) Toeplitz matrices. Our paper [2]
may be considered as a first step to the understanding of the asymptotic behavior
of individual inner eigenvalues of Toeplitz matrices. In the same vein, this paper
intends to understand the nature of individual eigenvectors as part of the whole,
independently of whether they are extreme or inner ones.
To state our main results, we need some notation. Let λ ∈ [0, M]. Then there
are uniquely defined ϕ1(λ) ∈ [0, ϕ0] and ϕ2(λ) ∈ [ϕ0 − 2π, 0] such that
g(ϕ1(λ)) = g(ϕ2(λ)) = λ;](https://image.slidesharecdn.com/1021278-250521173013-9c3fc6a2/85/Recent-Trends-In-Toeplitz-And-Pseudodifferential-Operators-Duduchava-R-28-320.jpg)
![18 A. Böttcher, S.M. Grudsky and E.A. Maksimenko
recall that g(x) := a(eix
). We put
ϕ(λ) =
ϕ1(λ) − ϕ2(λ)
2
, σ(λ) =
ϕ1(λ) + ϕ2(λ)
2
.
We have
a(z) − λ = z−r
arz2r
+ · · · + (a0 − λ)zr
+ · · · + a−r
= arz−r
2r
k=1
(z − zk(λ)),
and our assumptions imply that we can label the zeros zk(λ) so that the collection
Z(λ) of the zeros may be written as
{z1(λ), . . . , zr−1(λ), zr(λ), zr+1(λ), zr+2(λ), . . . , z2r(λ)}
= {u1(λ), . . . , ur−1(λ), eiϕ1(λ)
, eiϕ2(λ)
, 1/u1(λ), . . . , 1/ur−1(λ)} (1.4)
where |uν(λ)| 1 for 1 ≤ ν ≤ r − 1 and each uν(λ) depends continuously on
λ ∈ [0, M]. Here and in similar places below we write uk(λ) := uk(λ). We define
δ0 0 by
eδ0
= min
λ∈[0,M]
min
1≤ν≤r−1
|uν(λ)|.
Throughout the following, δ stands for any number in (0, δ0). Further, we denote
by hλ the function
hλ(z) =
r−1
ν=1
1 −
z
uν(λ)
.
The function Θ(λ) = hλ(eiϕ1(λ)
)/hλ(eiϕ2(λ)
) is continuous and nonzero on [0, M]
and we have Θ(0) = Θ(M) = 1. In [2], it was shown that the closed curve
[0, M] → C {0}, λ → Θ(λ)
has winding number zero. Let θ(λ) be the continuous argument of Θ(λ) for which
θ(0) = θ(M) = 0.
In [2], we proved that if n is large enough, then the function
fn : [0, M] → [0, (n + 1)π], fn(λ) = (n + 1)ϕ(λ) + θ(λ)
is bijective and increasing and that if λ
(n)
j,∗ is the unique solution of the equation
fn(λ
(n)
j,∗ ) = πj, then the eigenvalues λ
(n)
j satisfy
|λj − λ
(n)
j,∗ | ≤ K e−δn
for all j ∈ {1, . . . , n}, where K is a finite constant depending only on a. Thus, we
have
(n + 1)ϕ(λ
(n)
j ) + θ(λ
(n)
j ) = πj + O(e−δn
), (1.5)
uniformly in j ∈ {1, . . ., n}.](https://image.slidesharecdn.com/1021278-250521173013-9c3fc6a2/85/Recent-Trends-In-Toeplitz-And-Pseudodifferential-Operators-Duduchava-R-29-320.jpg)
![20 A. Böttcher, S.M. Grudsky and E.A. Maksimenko
Things can be a little simplified for symmetric matrices. Thus, suppose all
ak are real and ak = a−k for all k. We will show that then {u1(λ), . . . , ur−1(λ)} =
{u1(λ), . . . , ur−1(λ)}. Put
Qν(λ) =
|hλ(eiϕ(λ)
)| sin ϕ(λ)
(uν(λ) − eiϕ(λ))(uν(λ) − e−iϕ(λ))h
λ(uν(λ))
and let y
(n)
j (λ) = (y
(n)
j,m(λ))n
m=1 be given by
y
(n)
j,m(λ) = sin
mϕ(λ) +
θ(λ)
2
−
r−1
ν=1
Qν(λ)
1
uν(λ)m
+
(−1)j+1
uν(λ)n+1−m
. (1.6)
Theorem 1.3. Let n → ∞ and suppose λ
(n)
j ∈ (α, β). If ak = a−k for all k, then
y
(n)
j (λ
(n)
j )2
2 =
n
2
+ O(1)
uniformly in j, and the eigenvectors v
(n)
j are of the form
v
(n)
j = τ
(n)
j
y
(n)
j (λ
(n)
j )
y
(n)
j (λ
(n)
j )2
+ O∞(e−δn
)
where τ
(n)
j ∈ T and O∞(e−δn
) is as in the previous theorem.
Let J be the n × n matrix with ones on the counterdiagonal and zeros else-
where. Thus, (Jv)m = vn+1−m. A vector v is called symmetric if Jv = v and
skew-symmetric if Jv = −v. Trench [17] showed that the eigenvectors v
(n)
1 , v
(n)
3 , . . .
are all symmetric and that the eigenvectors v
(n)
2 , v
(n)
4 , . . . are all skew-symmetric.
From (1.5) we infer that
sin
(n + 1 − m)ϕ(λ
(n)
j ) +
θ(λ
(n)
j )
2
= (−1)j+1
sin
mϕ(λ
(n)
j ) +
θ(λ
(n)
j )
2
+ O(e−δn
)
and hence (1.6) implies that
(Jy
(n)
j (λ
(n)
j ))m = (−1)j+1
y
(n)
j,m(λ
(n)
j ) + O(e−δn
).
Consequently, apart from the term O(e−δn
), the vectors y
(n)
j (λ
(n)
j ) are symmetric
for j = 1, 3, . . . and skew-symmetric for j = 2, 4, . . .. This is in complete accordance
with Trench’s result.
Due to (1.5), we also have
sin
mϕ(λ
(n)
j ) +
θ(λ
(n)
j )
2
= sin
m −
n + 1
2
ϕ(λ
(n)
j )
+ O(e−δn
).](https://image.slidesharecdn.com/1021278-250521173013-9c3fc6a2/85/Recent-Trends-In-Toeplitz-And-Pseudodifferential-Operators-Duduchava-R-31-320.jpg)
![Eigenvectors of Large Hermitian Toeplitz Band Matrices 21
Thus, Theorem 1.3 remains valid with (1.6) replaced by
y
(n)
j,m(λ) = sin
m −
n + 1
2
ϕ(λ) +
πj
2
−
r−1
ν=1
Qν(λ)
1
uν(λ)m
+
(−1)j+1
uν(λ)n+1−m
. (1.7)
In this expression, the function θ has disappeared.
Define y
(n)
j again by (1.6). The following theorem in conjunction with Theo-
rem 1.3 proves (1.3).
Theorem 1.4. Let n → ∞ and suppose λ
(n)
j ∈ (α, β). If ak = a−k for all k, then
y
(n)
j (λ
(n)
j )
y
(n)
j (λ
(n)
j )2
− x
(n)
j
2
= O
j
n
.
The rest of the paper is as follows. We approach eigenvectors by using the
elementary observation that if λ is an eigenvalue of Tn(a), then every nonzero
column of the adjugate matrix of Tn(a) − λI = Tn(a − λ) is an eigenvector for λ.
In Section 2 we employ “exact” formulas by Trench and Widom for the inverse and
the determinant of a banded Toeplitz matrix to get a representation of the first
column of the adjugate matrix of Tn(a − λ) that will be convenient for asymptotic
analysis. This analysis is carried out in Section 3. On the basis of these results,
Theorems 1.1 and 1.2 are proved in Section 4, while the proofs of Theorems 1.3
and 1.4 are given in Section 4. Section 6 contains numerical results.
2. The first column of the adjugate matrix
The adjugate matrix adj B of an n × n matrix B = (bjk)n
j,k=1 is defined by
(adj B)jk = (−1)j+k
det Mkj
where Mkj is the (n − 1) × (n − 1) matrix that results from B by deleting the kth
row and the jth column. We have
(A − λI) adj (A − λI) = (det(A − λI))I.
Thus, if λ is an eigenvalue of A, then each nonzero column of adj (A − λI) is an
eigenvector. For an invertible matrix B,
adj B = (det B)B−1
. (2.1)
Formulas for det Tn(b) and T −1
n (b) were established by Widom [18] and Trench
[16], respectively. The purpose of this section is to transform Trench’s formula for
the first column of T −1
n (b) into a form that will be convenient for further analysis.](https://image.slidesharecdn.com/1021278-250521173013-9c3fc6a2/85/Recent-Trends-In-Toeplitz-And-Pseudodifferential-Operators-Duduchava-R-32-320.jpg)
![22 A. Böttcher, S.M. Grudsky and E.A. Maksimenko
Theorem 2.1. Let
b(t) =
q
k=−p
bktk
= bpt−q
p+q
j=1
(t − zj) (t ∈ T)
where p ≥ 1, q ≥ 1, bp = 0, and z1, . . . , zp+q are pairwise distinct nonzero complex
numbers. If n p + q and 1 ≤ m ≤ n, then the mth entry of the first column of
adj Tn(b) is
[adj Tn(b)]m,1 =
J⊂Z,|J|=p
CJ Wn
J
z∈J
Sm,J,z (2.2)
where Z = {z1, . . . , zp+q}, the sum is over all sets J ⊂ Z of cardinality p, and,
with J := Z J,
CJ =
z∈J
zq
z∈J,w∈J
1
z − w
, WJ = (−1)p
bp
z∈J
z,
Sm,J,z = −
1
bp
1
zm
w∈J{z}
1
z − w
.
Proof. It suffices to prove (2.2) under the assumption that det Tn(b) = 0 because
both sides of (2.2) are continuous functions of z1, . . . , zp+q. Thus, let det Tn(b) = 0.
We will employ (2.1) with B = Tn(b).
Trench [16] proved that [T −1
n (b)]m,1 equals
−
1
bp
D{1,...,p+q}(0, . . . , q − 1, q + n, . . . , q + n + p − 2, q + n − m)
D{1,...,p+q}(0, . . . , q − 1, q + n, . . . , q + n + p − 1)
(2.3)
where D{j1,...,jk}(s1, . . . , sk) denotes the determinant
det
⎛
⎜
⎜
⎜
⎝
zs1
j1
zs2
j1
. . . zsk
j1
zs1
j2
zs2
j2
. . . zsk
j2
.
.
.
.
.
.
.
.
.
zs1
jk
zs2
jk
. . . zsk
jk
⎞
⎟
⎟
⎟
⎠
.
Note that
DJ (s1 + ξ, . . . , sk + ξ) =
⎛
⎝
j∈J
zξ
j
⎞
⎠ DJ (s1, . . . , sk),
D{1,2,...,k}(0, 1, . . . , k − 1) =
j,∈J
j
(z − zj).](https://image.slidesharecdn.com/1021278-250521173013-9c3fc6a2/85/Recent-Trends-In-Toeplitz-And-Pseudodifferential-Operators-Duduchava-R-33-320.jpg)
![Eigenvectors of Large Hermitian Toeplitz Band Matrices 23
We first consider the denominator of (2.3). Put Z = {1, . . ., p + q}. Laplace
expansion along the last p columns gives
DZ (0, . . . , q − 1, q + n, . . . , q + n + p − 1)
=
J⊂Z,|J|=p
(−1)inv(J,J)
DJ (q + n, . . . , q + n + p − 1)DJ (0, . . . , q − 1)
=
J⊂Z,|J|=p
(−1)inv(J,J)
k∈J
zq+n
k
k,∈J
k
(z − zk)
k,∈J
k
(z − zk),
where inv(J, J) is the number of inversions in the permutation of length p + q
whose first q elements are the elements of the set J in increasing order and whose
last p elements are the elements of the set J in increasing order. A little thought
reveals that inv(J, J) is just the number of pairs (k, ) with k ∈ J, ∈ J, k .
We have
j∈J,s∈J
(zj − zs) =
∈J,k∈J
k
(z − zk)
k∈J,∈J
k
(zk − z)
= (−1)inv(J,J)
∈J,k∈J
k
(z − zk)
∈J,k∈J
k
(z − zk) (2.4)
and hence the denominator is equal to
Rn
J⊂Z,|J|=p
CJ Wn
J with Rn :=
(−1)pn
bn
p k
(z − zk).
A formula by Widom [18], which can also be found in [1], says that
det Tn(b) =
J⊂Z,|J|=p
CJ Wn
J .
Consequently, the denominator of (2.3) is nothing but Rn det Tn(b).
Let us now turn to the numerator of (2.3). This time Laplace expansion along
the last p columns yields
DZ(0, . . . , q − 1, q + n, . . . , q + n + p − 2, q + n − m)
=
J⊂Z,|J|=p
(−1)inv(J,J)
DJ (q + n, . . . , q + n + p − 1, q + n − m)DJ (0, . . . , q − 1)
=
J⊂Z,|J|=p
(−1)inv(J,J)
DJ (0, . . . , q − 1)
⎛
⎝
j∈J
zq+n
j
⎞
⎠ DJ (0, . . . , p − 2, −m).
Expanding DJ (0, . . . , p − 2, −m) by its last column we get
DJ (0, . . . , p − 2, −m) =
j∈J
(−1)inv(J{j},j)
z−m
j DJ{j}(0, . . . , p − 2) (2.5)](https://image.slidesharecdn.com/1021278-250521173013-9c3fc6a2/85/Recent-Trends-In-Toeplitz-And-Pseudodifferential-Operators-Duduchava-R-34-320.jpg)
![24 A. Böttcher, S.M. Grudsky and E.A. Maksimenko
with inv(J {j}, j) being the number of s ∈ J {j} such that s j. Thus, (2.5) is
j∈J
(−1)inv(J{j},j)
z−m
j
k,∈J{j}
k
(z − zk)
=
j∈J
z−m
j
k,∈J
k
(z − zk)
s∈J{j}
1
zj − zs
.
This in conjunction with (2.4) shows that the numerator of (2.2) equals
−bpRn
J⊂Z,|J|=p
CJ Wn
J
z∈J
Sm,J,z.
In summary, from (2.3) we obtain that
[T −1
n (b)]m,1 =
1
det Tn(b)
J⊂Z,|J|=p
CJ Wn
J
z∈J
Sm,J,z,
which after multiplication by det Tn(b) becomes (2.2).
3. The main terms of the first column
We now apply Theorem 2.1 to
b(t) = a(t) − λ = art−r
2r
k=1
(t − zk(λ)) (3.1)
where λ ∈ (α, β). The set Z = Z(λ) is given by (1.4). Let
d0(λ) = (−1)r
areiσ(λ)
r−1
k=1
uk(λ). (3.2)
In [2], we showed that d0(λ) 0 for all λ ∈ (0, M). The dependence on λ will
henceforth frequently be suppressed in notation. Let
J1 = {u1, . . . , ur−1, eiϕ1
}, J2 = {u1, . . . , ur−1, eiϕ2
}
and for ν ∈ {1, . . . , r − 1}, put
J0
ν = {u1, . . . , ur−1, 1/uν}.
Lemma 3.1. If J ⊂ Z, |J| = r, J /
∈ {J1, J2, J0
1 , . . . , J0
r−1}, then
|CJ Wn
j Sm,J,z| ≤ K
dn
0
sin ϕ
e−δn
for all z ∈ J, n ≥ 1, 1 ≤ m ≤ n, λ ∈ (α, β) with some finite constant K that does
not depend on z, n, m, λ.](https://image.slidesharecdn.com/1021278-250521173013-9c3fc6a2/85/Recent-Trends-In-Toeplitz-And-Pseudodifferential-Operators-Duduchava-R-35-320.jpg)
![Eigenvectors of Large Hermitian Toeplitz Band Matrices 25
Proof. If both eiϕ1
and eiϕ2
belong to J, then
J = {uν1 , . . . , uνk
, eiϕ1
, eiϕ2
, 1/us1 , . . . , 1/us
}
with k + = r − 2. Since
min
λ∈[α,β]
min
j1=j2
|uj1 (λ) − uj2 (λ)| 0,
we conclude that |CJ | ≤ K1. Here and in the following Ki denotes a finite constant
that is independent of λ ∈ [α, β]. We have k ≤ r − 2 and thus
|WJ | = |ar|
|uν1 . . . uνk
|
|us1 . . . us
|
≤
d0e−δ
|us1 . . . us
|
. (3.3)
If z ∈ {uν1 , . . . , uνk
, eiϕ1
, eiϕ2
}, then obviously |Sm,J,z| ≤ K2/ sin ϕ and hence
|CJ Wn
J Sm,J,z| ≤ K1K2
dn
0 e−δn
sin ϕ
.
In case z ∈ {1/us1 , . . . , 1/us
}, say z = 1/us1 , we have |Sm,J,z| ≤ K3|uν1 |m
, which
gives
|CJ Wn
J Sm,J,z| ≤ K1K3dn
0 e−δn |uν1 |m
|uν1 |n
≤ K1K3dn
0 e−δn
≤ K1K3
dn
0 e−δn
sin ϕ
.
The only other possibility for J is to be of the type
J = {uν1 , . . . , uνk
, eiϕ1
, 1/us1 , . . . , 1/us
}
with k + ≤ r − 1, k ≤ r − 2, ≥ 1. (The case where eiϕ1
is replaced by
eiϕ2
is completely analogous.) This time, |CJ | ≤ K4/ sin ϕ and (3.3) holds again.
For z ∈ {uν1 , . . . , uνk
, eiϕ1
} we have |Sm,J,z| ≤ K5 and thus get the assertion. If
z = 1/us for some s ∈ {s1, . . . , s}, say s = s1, then |Sm,J,z| ≤ K6|us1 |m
, and the
assertion follows as above, too.
Let
d1(λ) =
1
|hλ(eiϕ1(λ))hλ(eiϕ2(λ))|
r−1
k,s=1
1 −
1
uk(λ)us(λ)
−1
.
It is easily seen that d1(λ) 0 for all λ ∈ [0, M].
Lemma 3.2. If λ = λ
(n)
j ∈ (0, M), then
CJ1 Wn
J1
Sm,J1,eiϕ1 =
d1dn−1
0
sin ϕ
(−1)j
Ae−imϕ1
+ O(e−δn
)
,
CJ2 Wn
J2
Sm,J2,eiϕ2 =
d1dn−1
0
sin ϕ
(−1)j+1
Be−imϕ2
+ O(e−δn
)
uniformly in m and λ.](https://image.slidesharecdn.com/1021278-250521173013-9c3fc6a2/85/Recent-Trends-In-Toeplitz-And-Pseudodifferential-Operators-Duduchava-R-36-320.jpg)
![Eigenvectors of Large Hermitian Toeplitz Band Matrices 29
Proof. This time
Sm,J0
ν ,uk
= −
1
ar
1
um
k
1
(uk − 1/uν)
s=k(uk − us)
=
u−m
k
(−1)r−1aruk
1 − 1
ukuν
s=k us
s=k
1 − uk
us
= −
1
d0um
k
1
1 − 1
ukuν
s=k
1 − uk
us
.
Expressions for CJ0
ν
and WJ0
ν
were given in the proof of Lemma 3.4. It follows that
CJ0
ν
Wn
J0
ν
Sm,J0
ν ,uk
= Gν,k
d1dn−1
0
sin ϕ
1
un+1
ν um
k
where Gν,k equals
e2iσ
e−inσ
|h(eiϕ1
)h(eiϕ2
)| sin ϕ
(uν − e−iϕ1 )(uν − e−iϕ2 )h(uν)h(eiϕ1 )h(eiϕ2 )
s=k
1 − 1
usuν
s=k
1 − uk
us
.
Since
h(uν) = −
1
uν
s=ν
1 −
uν
us
,
we see that Gν,k remains bounded on [α, β]. Finally,
1
|un+1
ν um
k |
≤
1
|uν|n
≤ e−δn
.
Corollary 3.6. If λ = λ
(n)
j ∈ (α, β), then
[adj Tn(a − λ)]m,1 = (−1)j d1(λ)dn−1
0 (λ)
sin ϕ(λ)
[wj,m(λ) + O(e−δn
)]
uniformly in m and λ.
Proof. This follows from Theorem 2.1 and Lemmas 3.1 to 3.5 along with the fact
that d1 is bounded and bounded away from zero on [α, β].
4. The asymptotics of the eigenvectors
We now prove Theorem 1.1. There is a finite constant K1 such that |Dν| ≤ K1
and |Fν| ≤ K1 for all ν and all λ ∈ (α, β). Thus, summing up two finite geometric
series, we get
n
m=1
Dν
1
um
ν
+ Fν
(−1)j+1
e−i(n+1)σ
un+1−m
ν
2
≤ 2K2
1
1
|uν|2
1 − 1/|uν|2(n+1)
1 − 1/|uν|2
≤ K2](https://image.slidesharecdn.com/1021278-250521173013-9c3fc6a2/85/Recent-Trends-In-Toeplitz-And-Pseudodifferential-Operators-Duduchava-R-40-320.jpg)
![30 A. Böttcher, S.M. Grudsky and E.A. Maksimenko
for all ν, n, λ. We further have
n
m=1
Ae−imϕ1
− Be−imϕ2
2
=
n
m=1
e−imϕ1
2h(eiϕ1 )
−
e−imϕ2
2h(eiϕ2 )
2
=
n
m=1
1
4|h(eiϕ1 )|2
+
1
4|h(eiϕ1 )|2
−
n
m=1
e−2imϕ
4h(eiϕ1 )h(eiϕ2 )
+
e2imϕ
4h(eiϕ1 )h(eiϕ2 )
.
The first sum is of the form
n
m=1(γ/4) and therefore equals (n/4)γ. Hence,
because of (3.4) we are left to prove that
n
m=1
eiθ(λ
(n)
j )
e2imϕ(λ
(n)
j )
≤ K3 (4.1)
for all n and j such that λ
(n)
j ∈ (α, β). The sum in (4.1) is
ei[(n+1)ϕ(λ
(n)
j )+θ(λ
(n)
j )]
sin nϕ(λ
(n)
j )
sin ϕ(λ
(n)
j )
.
Thus, (4.1) will follow as soon as we have shown that
sin nϕ(λ
(n)
j )
sin ϕ(λ
(n)
j )
≤ K3
for all n and j in question. From (1.5) we infer that
nϕ(λ
(n)
j ) = πj − ϕ(λ
(n)
j ) − θ(λ
(n)
j ) + O(e−δn
),
which implies that
sin nϕ(λ
(n)
j ) = (−1)j+1
sin
ϕ(λ
(n)
j ) + θ(λ
(n)
j ) + O(e−δn
).
Suppose first that 0 ϕ(λ
(n)
j ) ≤ π/2. Then
sin
ϕ(λ
(n)
j ) + θ(λ
(n)
j )
sin ϕ(λ
(n)
j )
≤
π
2
|ϕ(λ
(n)
j ) + θ(λ
(n)
j )|
|ϕ(λ
(n)
j )|
≤
π
2
1 +
|θ(λ
(n)
j )|
|ϕ(λ
(n)
j )|
. (4.2)
In [2], we proved that |θ/ϕ| is bounded on (0, M). Thus, the right-hand side of
(4.2) is bounded by some K3 for all n and j. If π/2 ϕ(λ
(n)
j ) π, we may replace
(4.2) by the upper bound
π
2
1 +
|θ(λ
(n)
j )|
|π − ϕ(λ
(n)
j )|
.
We know again from [2] that |θ/(π −ϕ)| is bounded on (0, M). This completes the
proof of Theorem 1.1.](https://image.slidesharecdn.com/1021278-250521173013-9c3fc6a2/85/Recent-Trends-In-Toeplitz-And-Pseudodifferential-Operators-Duduchava-R-41-320.jpg)
![Eigenvectors of Large Hermitian Toeplitz Band Matrices 31
Here is the proof of Theorem 1.2. By virtue of Theorem 1.1, wj(λ
(n)
j )2 1
whenever n is sufficiently large. Corollary 3.6 therefore implies that the first column
of adjTn(a − λ
(n)
j ) is nonzero and thus an eigenvector for λ
(n)
j for all n ≥ n0 and
all 1 ≤ j ≤ n such that λ
(n)
j ∈ (α, β). Again by Corollary 3.6, the mth entry of
this column is
d1(λ)dn−1
0 (λ)
sin ϕ(λ)
[wj,m(λ) + ξ
(n)
j,m]
λ=λ
(n)
j
where |ξ
(n)
j,m| ≤ Ke−δn
for all n and j under consideration and K does not depend
on m, n, j. It follows that
wj(λ
(n)
j ) +
ξ
(n)
j,m
n
m=1
= wj(λ
(n)
j ) + O∞(e−δn
)
is also an eigenvector for λ
(n)
j . Consequently,
wj(λ
(n)
j ) + O∞(e−δn
)
wj(λ
(n)
j ) + O∞(e−δn)2
=
wj(λ
(n)
j )
wj(λ
(n)
j )2
+ O∞(e−δn
) (4.3)
is a normalized eigenvector for λ
(n)
j . From (1.5) we deduce that all eigenvalues
of Tn(a) are simple. Thus, v
(n)
j is a scalar multiple of modulus 1 of (4.3). This
completes the proof of Theorem 1.2.
5. Symmetric matrices
The matrices Tn(a) are all symmetric if and only if all ak are real and ak = a−k for
all k. Obviously, this is equivalent to the requirement that the real-valued function
g(x) := a(eix
) be even, that is, g(x) = g(−x) for all x. Thus, suppose g is even. In
that case
ϕ0 = π, ϕ1(λ) = −ϕ2(λ) = ϕ(λ), σ(λ) = 0.
Moreover, for t ∈ T we have
art−r
2r
k=1
(t − zk(λ)) = a(t) − λ = a(1/t) − λ
= artr
2r
k=1
(1/t − zk(λ)) = ar
2r
k=1
zk(λ)
t−r
2r
k=1
(t − 1/zk(λ)),
which in conjunction with (1.4) implies that
{u1(λ), . . . , ur−1(λ)} = {u1(λ), . . . , ur−1(λ)}. (5.1)
The coefficients of the polynomial hλ(t) are symmetric functions of
1/u1(λ), . . . , 1/ur−1(λ).](https://image.slidesharecdn.com/1021278-250521173013-9c3fc6a2/85/Recent-Trends-In-Toeplitz-And-Pseudodifferential-Operators-Duduchava-R-42-320.jpg)
![34 A. Böttcher, S.M. Grudsky and E.A. Maksimenko
that is,
sin
mϕ(λ
(n)
j ) + θ(λ
(n)
j ) − sin
mjπ
n + 1
n
m=1 2
= O
j
√
n
. (5.4)
Combining (5.3) and (5.4)we obtain that
y
(n)
j −
n + 1
2
x
(n)
j
2
= O
j
n
+ O
j
√
n
= O
j
√
n
, (5.5)
which implies in particular that
y
(n)
j 2 =
n + 1
2
1 + O
j
n
. (5.6)
Clearly, estimates (5.5) and (5.6) yield the asserted estimate. This completes the
proof of Theorem 1.4.
6. Numerical results
Given Tn(a), determine the approximate eigenvalue λ
(n)
j,∗ from the equation
(n + 1)ϕ(λ
(n)
j,∗ ) + θ(λ
(n)
j,∗ ) = πj.
In [2], we proposed an exponentially fast iteration method for solving this equation.
Let w
(n)
j (λ) ∈ Cn
be as in Section 1 and put
w
(n)
j,∗ =
w
(n)
j (λ
(n)
j,∗ )
w
(n)
j (λ
(n)
j,∗ )2
.
We define the distance between the normalized eigenvector v
(n)
j and the normalized
vector w
(n)
j,∗ by
(v
(n)
j , w
(n)
j,∗ ) := min
τ∈T
τv
(n)
j − w
(n)
j,∗ 2 = 2 − 2 v
(n)
j , w
(n)
j,∗
and put
Δ
(n)
∗ = max
1≤j≤n
|λ
(n)
j − λ
(n)
j,∗ |,
Δ(n)
v,w = max
1≤j≤n
(v
(n)
j , w
(n)
j,∗ ),
Δ(n)
r = max
1≤j≤n
Tn(a)w
(n)
j,∗ ) − λ
(n)
j,∗ w
(n)
j,∗ 2.
The tables following below show these errors for three concrete choices of the
generating function a.](https://image.slidesharecdn.com/1021278-250521173013-9c3fc6a2/85/Recent-Trends-In-Toeplitz-And-Pseudodifferential-Operators-Duduchava-R-45-320.jpg)
![Eigenvectors of Large Hermitian Toeplitz Band Matrices 35
For a(t) = 8 − 5t − 5t−1
+ t2
+ t−2
we have
n = 10 n = 20 n = 50 n = 100 n = 150
Δ
(n)
∗ 5.4 · 10−7
1.1 · 10−11
5.2 · 10−25
1.7 · 10−46
9.6 · 10−68
Δ
(n)
v,w 2.0 · 10−6
1.1 · 10−10
2.0 · 10−23
1.9 · 10−44
2.0 · 10−65
Δ
(n)
r 8.0 · 10−6
2.7 · 10−10
3.4 · 10−23
2.2 · 10−44
1.9 · 10−65
If a(t) = 8 + (−4 − 2i)t + (−4 − 2i)t−1
+ it − it−1
then
n = 10 n = 20 n = 50 n = 100 n = 150
Δ
(n)
∗ 3.8 · 10−8
2.8 · 10−13
2.9 · 10−30
5.9 · 10−58
1.6 · 10−85
Δ
(n)
v,w 1.8 · 10−7
4.7 · 10−13
2.0 · 10−29
7.0 · 10−57
2.4 · 10−84
Δ
(n)
r 5.4 · 10−7
1.3 · 10−12
2.7 · 10−29
6.7 · 10−57
1.9 · 10−84
In the case where a(t) = 24 + (−12 − 3i)t + (−12 + 3i)t−1
+ it3
− it−3
we get
n = 10 n = 20 n = 50 n = 100 n = 150
Δ
(n)
∗ 6.6 · 10−6
1.2 · 10−10
7.6 · 10−24
1.4 · 10−45
3.3 · 10−67
Δ
(n)
v,w 1.9 · 10−6
1.3 · 10−10
2.0 · 10−23
7.2 · 10−45
2.8 · 10−66
Δ
(n)
r 2.5 · 10−5
8.6 · 10−10
7.3 · 10−23
1.9 · 10−44
5.9 · 10−66
References
[1] A. Böttcher and S.M. Grudsky, Spectral Properties of Banded Toeplitz Matrices,
SIAM, Philadelphia 2005.
[2] A. Böttcher, S.M. Grudsky, E.A. Maksimenko, Inside the eigenvalues of certain Her-
mitian Toeplitz band matrices. Journal of Computational and Applied Mathematics
233 (2010), 2245–2264.
[3] A. Böttcher, S.M. Grudsky, E.A. Maksimenko, and J. Unterberger, The first or-
der asymptotics of the extreme eigenvectors of certain Hermitian Toeplitz matrices.
Integral Equations and Operator Theory 63 (2009), 165–180.
[4] A. Böttcher, S.M. Grudsky, and J. Unterberger, Asymptotic pseudomodes of Toeplitz
matrices, Operators and Matrices 2 (2008), 525–541.
[5] A. Böttcher and B. Silbermann, Introduction to Large Truncated Toeplitz Matrices,
Universitext, Springer-Verlag, New York 1999.
[6] C. Estatico and S. Serra Capizzano, Superoptimal approximation for unbounded sym-
bols, Linear Algebra Appl. 428 (2008), 564–585.
[7] U. Grenander and G. Szegö, Toeplitz Forms and Their Applications, University of
California Press, Berkeley 1958.
[8] C.M. Hurvich and Yi Lu, On the complexity of the preconditioned conjugate gradient
algorithm for solving Toeplitz systems with a Fisher-Hartwig singularity, SIAM J.
Matrix Anal. Appl. 27 (2005), 638–653.
[9] M. Kac, W.L. Murdock, and G. Szegö, On the eigenvalues of certain Hermitian
forms, J. Rational Mech. Anal. 2 (1953), 767–800.](https://image.slidesharecdn.com/1021278-250521173013-9c3fc6a2/85/Recent-Trends-In-Toeplitz-And-Pseudodifferential-Operators-Duduchava-R-46-320.jpg)
![36 A. Böttcher, S.M. Grudsky and E.A. Maksimenko
[10] A.Yu. Novosel’tsev and I.B. Simonenko, Dependence of the asymptotics of extreme
eigenvalues of truncated Toeplitz matrices on the rate of attaining an extremum by
a symbol, St. Petersburg Math. J. 16 (2005), 713–718.
[11] S.V. Parter, Extreme eigenvalues of Toeplitz forms and applications to elliptic dif-
ference equations, Trans. Amer. Math. Soc. 99 (1961), 153–192.
[12] S.V. Parter, On the extreme eigenvalues of Toeplitz matrices, Trans. Amer. Math.
Soc. 100 (1961), 263–276.
[13] S. Serra Capizzano, On the extreme spectral properties of Toeplitz matrices generated
by L1
functions with several minima/maxima, BIT 36 (1996), 135–142.
[14] S. Serra Capizzano, On the extreme eigenvalues of Hermitian (block ) Toeplitz ma-
trices, Linear Algebra Appl. 270 (1998), 109–129.
[15] S. Serra Capizzano and P. Tilli, Extreme singular values and eigenvalues of non-
Hermitian block Toeplitz matrices, J. Comput. Appl. Math. 108 (1999), 113–130.
[16] W.F. Trench, Explicit inversion formulas for Toeplitz band matrices, SIAM J. Al-
gebraic Discrete Methods 6 (1985), 546–554.
[17] W.F. Trench, Interlacement of the even and odd spectra of real symmetric Toeplitz
matrices, Linear Algebra Appl. 195 (1993), 59–68.
[18] H. Widom, On the eigenvalues of certain Hermitian operators, Trans. Amer. Math.
Soc. 88 (1958), 491–522.
[19] N.L. Zamarashkin and E.E. Tyrtyshnikov, Distribution of the eigenvalues and sin-
gular numbers of Toeplitz matrices under weakened requirements on the generating
function, Sb. Math. 188 (1997), 1191–1201.
[20] P. Zizler, R.A. Zuidwijk, K.F. Taylor, and S. Arimoto, A finer aspect of eigenvalue
distribution of selfadjoint band Toeplitz matrices, SIAM J. Matrix Anal. Appl. 24
(2002), 59–67.
Albrecht Böttcher
Fakultät für Mathematik
TU Chemnitz
D-09107 Chemnitz, Germany
e-mail: aboettch@mathematik.tu-chemnitz.de
Sergei M. Grudsky
Departamento de Matemáticas, CINVESTAV del I.P.N.
Apartado Postal 14-740
07000 México, D.F., México
e-mail: grudsky@math.cinvestav.mx
Egor A. Maksimenko
Departamento de Matemáticas, CINVESTAV del I.P.N.
Apartado Postal 14-740
07000 México, D.F., México
e-mail: emaximen@math.cinvestav.mx](https://image.slidesharecdn.com/1021278-250521173013-9c3fc6a2/85/Recent-Trends-In-Toeplitz-And-Pseudodifferential-Operators-Duduchava-R-47-320.jpg)
![Operator Theory:
Advances and Applications, Vol. 210, 37–42
c
2010 Springer Basel AG
Complete Quasi-wandering Sets
and Kernels of Functional Operators
Victor D. Didenko
To Nikolai Vasilevski on the occasion of his 60th birthday
Abstract. Kernels of functional operators generated by mapping that possess
complete quasi-wandering sets are studied. It is shown that the kernels of the
operators under consideration either consist of a zero element or contain a
subset isomorphic to a space L∞(S), where S ⊂ Rn
has a positive Lebesgue
measure. Consequently, such operators are Fredholm if and only if they are
invertible.
Mathematics Subject Classification (2000). Primary 47B33, 39A33, 45E10; Sec-
ondary 42C40, 39B42.
Keywords. Quasi-wandering set, homogeneous equation, solution.
1. Introduction
Let X be a domain in Rn
provided with the Lebesgue measure μ, and let α : X →
X be a measurable mapping satisfying the compatibility condition. Thus if E is a
Lebesgue measurable subset of X and if μ(E) = 0, then μ(α−1
(E)) = 0. Assume
that A0, A1, . . . , As : X → Cd×d
are matrix functions, and consider the functional
equation
A0(x)ϕ(x) + A1(x)ϕ(α(x)) + · · · + As(x)ϕ(αs
(x)) = f(x) (1.1)
where ϕ : X → Cd
is an unknown vector function and f : X → Cd
is a given
vector function. Equations of the form (1.1) arise in various fields of mathematics
and its applications, and there is vast literature where different properties of such
equations and the corresponding operators are studied. For details, the reader can
consult [1, 2, 4, 5, 9, 10, 12] and literature therein.
Let us mention a few functional operators relevant to our following discussion.
Let A = A(x), x ∈ Rn
be a matrix function and M ∈ Rn×n
a non-singular integer
expansive matrix – i.e., all eigenvalues λj, j = 1, 2, . . ., n of the matrix M satisfy](https://image.slidesharecdn.com/1021278-250521173013-9c3fc6a2/85/Recent-Trends-In-Toeplitz-And-Pseudodifferential-Operators-Duduchava-R-48-320.jpg)
![38 V.D. Didenko
the inequality |λj| 1. Equations of the form
ϕ(x) = A(x)ϕ(Mx) (1.2)
appear in most publications concerning wavelets or their applications. In particu-
lar, equation (1.2) arises in discussions of the discrete refinement equation
ψ(x) =
k∈Zn
akψ(Mx − k), x ∈ Rn
, (1.3)
or the continuous refinement equation
ψ(x) =
Rn
c(x − My)ψ(x) dy, x ∈ Rn
. (1.4)
Note that the refinement equation (1.3) is associated with the functional equa-
tion (1.2), where the matrix A has a uniformly convergent Fourier series; whereas
equation (1.4) leads to the equation (1.2), with the matrix A that is the Fourier
image of the matrix c ∈ Ld×d
1 (Rn
).
Let us also recall another class of functional equations. Consider X = Rn
and
choose an h ∈ Rn
, h = 0. The difference equations
A0(x)ϕ(x) + A1(x)ϕ(x + h) + · · · + As(x)ϕ(x + sh) = f(x),
that often arise in applications have been investigated by various authors.
In the present paper, the kernel spaces and the Fredholm properties of the
equations mentioned are studied from a unified point of view.
Let
Uα :=
s
j=0
AjT j
α (1.5)
be the operator defined by the left-hand side of equation (1.1), where
Tαϕ(x) := ϕ(α(x)),
and Aj, j = 0, 1, . . . , s are the operators of multiplication by the matrices Aj(x).
In the present paper, the operator Uα is considered on the space Ld
2(X) of all
Lebesgue measurable square summable vector functions. We study the kernel space
of the functional operators for mappings α that possess the so-called complete
quasi-wandering set. For such mappings, the kernel space of the above-mentioned
operator Uα has a distinctive property – viz. it either consists of the single element
ϕ0 = 0 or it contains a subset isomorphic to a space L∞(S), where S is a subset of
Rn
with a positive Lebesgue measure. Note that it was conjectured in [3] that the
kernel of any operator (1.5) on Lp space is either infinite-dimensional or only has
the zero element. However, an example from [11] shows that this conjecture is not
true in general. On the other hand, the problem mentioned is closely connected to
the hypothesis that any operator (1.5) is Fredholm if and only if it is invertible.
Although valid for a number of operator algebras [1, 8], this hypothesis is also not
true in general [6]. Thus, this paper presents another class of operators when both
hypotheses are true. Moreover, our approach shows that the kernel space of the
operators under consideration can include subspaces as ‘massive’ as L∞(S).](https://image.slidesharecdn.com/1021278-250521173013-9c3fc6a2/85/Recent-Trends-In-Toeplitz-And-Pseudodifferential-Operators-Duduchava-R-49-320.jpg)
![Complete Quasi-wandering Sets 39
2. The kernel space of the operator Uα
Now let us assume that the mapping α : X → X is invertible, and the inverse
mapping α−1
also satisfies the compatibility condition.
Definition 2.1. A Lebesgue measurable set E ⊂ X is a quasi-wandering set for the
mapping α if, for any j, k ∈ Z, j = k, either αj
(E) ∩ αk
(E) = ∅, or if αj1
(E) ∩
αk1
(E) = ∅ for some j1, k1 ∈ Z, then αj1
(x) = αk1
(x) for all x ∈ E.
If αj
(E) ∩ αk
(E) = ∅ for all j, k ∈ Z, j = k, the set E is called the wandering
set for α.
Definition 2.2. A quasi-wandering set E is called complete if
X =
!
j∈Z
αj
(E).
It is clear that if a mapping α : X → X possesses a complete quasi-wandering
set E and for some indices j, k, j = k, αj
(x) = αk
(x) for all x ∈ E, then
X =
N
!
j=0
αj
(E)
for an N ∈ N.
Let us now consider a few mappings with complete quasi-wandering sets.
Example 1. Suppose X = R and α : R → R is the shift operator
α(x) := x − 1.
The mapping α satisfies all of the conditions mentioned. It is invertible and the
interval [0, 1) is a complete wandering set for α.
Example 2. Let X = R2
, and let Λ be a diagonal matrix
Λ =
λ1 0
0 λ2
,
where λ1 = 0 and λ2 1. Consider the mapping
αΛ(x) := Λx.
Then the set
EΛ :=
(x, y) ∈ R2
, x ∈ R and y ∈ [−1, −1/λ2) ∪ (1/λ2, 1]
#
is a complete wandering set for the mapping αΛ.
Example 3. Let X = R3
, m ∈ N, m ≥ and let
αR(x) := Rx,
where R is the rotation matrix
R =
⎛
⎝
1 0 0
0 cos(π/m) − sin(π/m)
0 sin(π/m) cos(π/m)
⎞
⎠ .](https://image.slidesharecdn.com/1021278-250521173013-9c3fc6a2/85/Recent-Trends-In-Toeplitz-And-Pseudodifferential-Operators-Duduchava-R-50-320.jpg)
![Complete Quasi-wandering Sets 41
restriction of $
m on the set Sα is a non-zero element of L∞(Sα), then
X
| $
m(x)ϕ0(x)|2
dx =
∪j∈Zαj (Eα)
| $
m(x)ϕ0(x)|2
dx ≥
αj0 (Eα)
| $
m(x)ϕ0(x)|2
dx
≥
Sα
| $
m(x)ϕ0(x)|2
dx ≥ ε2
Sα
| $
m(x)|2
dx 0.
Thus the kernel of the operator Uα contains the subset
S := { $
mϕ0 : m ∈ L∞(Sα)}
and the proof is complete.
The connection between some sets related to the operators of multiplication
by expansive matrices M and the kernel spaces of the corresponding refinement
operators was first noted in [7]. It turns out that these relations have a universal
nature and can be extended to general mappings α having quasi-wandering sets.
This allows us to characterize the Fredholm properties of the functional operators.
Corollary 2.4. Let Uα : L2(X) → L2(X) be a Φ+
-operator. If the mapping α :
X → X possesses a complete quasi-wandering set, then
ker Uα = 0.
Now consider functional operators generated by diffeomorphisms.
Theorem 2.5. Let A1, A2, . . . , As ∈ Ld×d
∞ (X) and let α : X → X be a differentiable
mapping such that the Jacobian Jα of α satisfies the inequality
0 r1 ≤ Jα(x) ≤ r2, r1, r2 ∈ R (2.2)
for all x ∈ X. If the mapping α has a complete quasi-wandering set, then the
operator Uα : Ld
2(X) → Ld
2(X) is Fredholm if and only if it is invertible.
Proof. If α has a complete quasi-wandering set E, then E is also a complete quasi-
wandering set for the inverse mapping α−1
. The proof of Theorem 2.5 therefore
follows from Theorem 2.3 and from condition (2.2), which implies that the struc-
ture of the adjoint operator U∗
α is similar to the structure of the operator Uα.
Remark 2.6. This result is known for some classes of the functional operators [1, 8].
One way to prove it is to show that the algebra generated by the corresponding
operator Tα contains no non-trivial compact operators – see [1, Theorem 8.3].
On the other hand, the approach used here allows us to obtain certain additional
information concerning the kernels of the operators under consideration.
Remark 2.7. In theory of dynamical systems, the systems with complete wander-
ing sets are called complete dissipative systems [13]. Thus, complete dissipative
systems are Fredholm if and only if they are invertible.
Acknowledgment
The author would like to thank Alexei Karlovich for pointing out reference [6].](https://image.slidesharecdn.com/1021278-250521173013-9c3fc6a2/85/Recent-Trends-In-Toeplitz-And-Pseudodifferential-Operators-Duduchava-R-52-320.jpg)
![42 V.D. Didenko
References
[1] A. Antonevich, Linear Functional Equations. Operator Approach. Operator Theory.
Advances and Applications, Vol. 83, Birkhäuser, Basel-Boston-Berlin, 1996.
[2] A. Antonevich, A. Lebedev, Functional Differential Equations: C∗
-Theory. Pitman
Monographs and Surveys in Pure and Applied Mathematics, Vol. 70. Longman,
Harlow, 1994.
[3] N.V. Azbelev, G.G. Islamov, A certain class of functional-differential equations. Dif-
ferencial’nye Uravnenija, 12 (1976), 417–427.
[4] G. Belitskii, V. Tkachenko, One-dimensional functional equations. Operator Theory:
Advances and Applications, Vol. 144, Birkhäuser, Basel, 2003.
[5] E. Castillo, A. Iglesias, R. Ruı́z-Cobo, Functional equations in applied sciences.
Mathematics in Science and Engineering, Vol. 199, Elsevier, Amsterdam, 2005.
[6] A.V. Chistyakov, A pathological counterexample to the non-Fredholmness conjecture
in algebras of weighted shift operators, Izv. Vyssh. Uchebn. Zaved. Mat., No. 10
(1995), 76–86.
[7] V.D. Didenko, Fredholm properties of multivariate refinement equations. Math.
Meth. Appl. Sci., 30 (2007), 1639–1644
[8] Yu.I. Karlovich, V.G. Kravchenko, G.S. Litvinchuk, Invertibility of functional oper-
ators in Banach spaces. In: Functional-differential equations, Perm. Politekh. Inst.,
Perm’, 1990, 18–58.
[9] V.G. Kravchenko, G.S. Litvinchuk, Introduction to the theory of singular integral
operators with shift. Mathematics and its Applications, Vol. 289, Kluwer, Dordrecht,
1994.
[10] M. Kuczma, Functional equations in single variable. Monografie Matematyczne, Vol.
46, PWN, Warszawa, 1968.
[11] V.G. Kurbatov, A conjecture in the theory of functional-differential equations, Dif-
ferentsial’nye Uravnenija, 14 (1978), 2074–2075.
[12] G.S. Litvinchuk, Solvability theory of boundary value problems and singular integral
equations with shift. Mathematics and its Applications, Vol. 523, Kluwer, Dordrecht,
2000.
[13] P.J. Nicholls, The Ergodic Theory of Discrete Groups. London Mathematical Society
Lecture Note Series, Vol. 143, Cambridge University press, Cambridge, 1989.
Victor D. Didenko
Department of Mathematics
University Brunei Darussalam
Bandar Seri Begawan
BE1410 Brunei
e-mail: diviol@gmail.com](https://image.slidesharecdn.com/1021278-250521173013-9c3fc6a2/85/Recent-Trends-In-Toeplitz-And-Pseudodifferential-Operators-Duduchava-R-53-320.jpg)
![Operator Theory:
Advances and Applications, Vol. 210, 43–77
c
2010 Springer Basel AG
Lions’ Lemma, Korn’s Inequalities and
the Lamé Operator on Hypersurfaces
Roland Duduchava
Dedicated to my friend and colleague Nikolai Vasilevski
on the occasion of his 60th birthday anniversary
Abstract. We investigate partial differential equations on hypersurfaces writ-
ten in the Cartesian coordinates of the ambient space. In particular, we gen-
eralize essentially Lions’ Lemma, prove Korn’s inequality and establish the
unique continuation property from the boundary for Killing’s vector fields,
which are analogues of rigid motions in the Euclidean space. The obtained
results, the Lax-Milgram lemma and some other results are applied to the
investigation of the basic Dirichlet and Neumann boundary value problems
for the Lamé equation on a hypersurface.
Mathematics Subject Classification (2000). 35J57, 74J35, 58J32.
Keywords. Lions’s Lemma, Korn’s inequality, Killing’s fields, Lax-Milgram
lemma, Lamé equation, Boundary value problems.
Introduction
Partial differential equations (PDEs) on hypersurfaces and corresponding bound-
ary value problems (BVPs) appear rather often in applications: see [Ha1, §72]
for the heat conduction by surfaces, [Ar1, §10] for the equations of surface flow,
[Ci1], [Ci3],[Ci4], [Ko2], [Go1] for thin flexural shell problems in elasticity, [AC1]
for the vacuum Einstein equations describing gravitational fields, [TZ1, TW1] for
the Navier-Stokes equations on spherical domains and spheres, [MM1] for minimal
surfaces, [AMM1] for diffusion by surfaces, as well as the references therein. Fur-
thermore, such equations arise naturally while studying the asymptotic behavior of
solutions to elliptic boundary value problems in a neighborhood of conical points
(see the classical reference [Ko1]).
The investigation was supported by the grant of the Georgian National Science Foundation
GNSF/ST07/3-175.](https://image.slidesharecdn.com/1021278-250521173013-9c3fc6a2/85/Recent-Trends-In-Toeplitz-And-Pseudodifferential-Operators-Duduchava-R-54-320.jpg)
Recent Trends In Toeplitz And Pseudodifferential Operators Duduchava R
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- 6. Operator Theory: Advances and Applications Vol. 210 Founded in 1979 by Israel Gohberg Editors: Joseph A. Ball (Blacksburg, VA, USA) Harry Dym (Rehovot, Israel) Marinus A. Kaashoek (Amsterdam, The Netherlands) Heinz Langer (Vienna, Austria) Christiane Tretter (Bern, Switzerland) Associate Editors: Vadim Adamyan (Odessa, Ukraine) Albrecht Böttcher (Chemnitz, Germany) B. Malcolm Brown (Cardiff, UK) Raul Curto (Iowa City, IA, USA) Fritz Gesztesy (Columbia, MO, USA) Pavel Kurasov (Lund, Sweden) Leonid E. Lerer (Haifa, Israel) Vern Paulsen (Houston, TX, USA) Mihai Putinar (Santa Barbara, CA, USA) Leiba Rodman (Williamsburg, VI, USA) Ilya M. Spitkovsky (Williamsburg, VI, USA) Honorary and Advisory Editorial Board: 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, AB, Canada) Peter D. Lax (New York, NY, USA) Donald Sarason (Berkeley, CA, USA) Bernd Silbermann (Chemnitz, Germany) Harold Widom (Santa Cruz, CA, 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)
- 7. Recent Trends in Toeplitz and Pseudodifferential Operators The Nikolai Vasilevskii Anniversary Volume Roland Duduchava Israel Gohberg Sergei M. Grudsky Vladimir Rabinovich Editors Birkhäuser
- 8. 2010 Mathematics Subject Classification: 47-06 Library of Congress Control Number: 2010928340 Bibliographic information published by Die Deutsche Bibliothek. Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de ISBN 978-3-0346-0547-2 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2010 Springer Basel AG P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced from chlorine-free pulp. TCF∞ Printed in Germany ISBN 978-3-0346-0547-2 e-ISBN 978-3-0346-0548-9 9 8 7 6 5 4 3 2 1 www.birkhauser.ch Editors: Roland Duduchava Department of Mathematical Physics Andrea Razmadze Mathematical Institute M. Alexidze str. 1 Tbilisi 0193 Georgia e-mail: dudu@rmi.acnet.ge Sergei M. Grudsky Departamento de Matemáticas CINVESTAV Av. I.P.N. 2508 Col. San Pedro Zacatenco Apartado Postal14-740 Mexico,D.F. Mexico e-mail: grudsky@math.cinvestav.mx Israel Gohberg (Z”L) Vladimir Rabinovich National Polytechnic Institute ESIME Zacatenco Av. IPN 07738 Mexico, D.F. Mexico e-mail:vladimir.rabinovich@gmail.com
- 9. Contents S. Grudsky, Y. Latushkin and M. Shapiro The Life and Work of Nikolai Vasilevski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Principal publications of Nikolai Vasilevski . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 A. Böttcher, S.M. Grudsky an E.A. Maksimenko On the Structure of the Eigenvectors of Large Hermitian Toeplitz Band Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 V.D. Didenko Complete Quasi-wandering Sets and Kernels of Functional Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 R. Duduchava Lions’ Lemma, Korn’s Inequalities and the Lamé Operator on Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 J.O. González-Cervanves, M.E. Luna-Elizarrarás and M. Shapiro On the Bergman Theory for Solenoidal and Irrotational Vector Fields, I: General Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 M.G. Hajibayov and S.G. Samko Weighted Estimates of Generalized Potentials in Variable Exponent Lebesgue Spaces on Homogeneous Spaces . . . . . . . . . . . . . . . . . . 107 Yu.I. Karlovich and J. Loreto Hernández Wiener-Hopf Operators with Oscillating Symbols on Weighted Lebesgue Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 V.V. Kucherenko and A. Kryvko On the Ill-posed Hyperbolic Systems with a Multiplicity Change Point of Not Less Than the Third Order . . . . . . . . . . . . . . . . . . . . . 147 M. Loaiza and A. Sánchez-Nungaray On C∗ -Algebras of Super Toeplitz Operators with Radial Symbols . . . 175
- 10. vi Contents H. Mascarenhas and B. Silbermann Universality of Some C∗ -Algebra Generated by a Unitary and a Self-adjoint Idempotent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 R. Quiroga-Barranco Commutative Algebras of Toeplitz Operators and Lagrangian Foliations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 V. Rabinovich and S. Roch Exponential Estimates of Eigenfunctions of Matrix Schrödinger and Dirac Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 N. Tarkhanov The Laplace-Beltrami Operator on a Rotationally Symmetric Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 A. Unterberger On the Structure of Operators with Automorphic Symbols . . . . . . . . . . . 233 H. Upmeier Hilbert Bundles and Flat Connexions over Hermitian Symmetric Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
- 12. Operator Theory: Advances and Applications, Vol. 210, 1–14 c 2010 Springer Basel AG The Life and Work of Nikolai Vasilevski Sergei Grudsky, Yuri Latushkin and Michael Shapiro Nikolai Leonidovich Vasilevski was born on January 21, 1948 in Odessa, Ukraine. His father, Leonid Semenovich Vasilevski, was a lecturer at Odessa Institute of Civil Engineering, his mother, Maria Nikolaevna Krivtsova, was a docent at the Department of Mathematics and Mechanics of Odessa State University. In 1966 Nikolai graduated from Odessa High School Number 116, a school with special emphasis in mathematics and physics, that made a big impact at his creative and active attitude not only to mathematics, but to life in general. It was a very selective high school accepting talented children from all over the city, and famous for a high quality selection of teachers. A creative, nonstandard, and at the same time highly personal approach to teaching was combined at the school with a demanding attitude towards students. His mathematics instructor at the high school was Tatjana Aleksandrovna Shevchenko, a talented and dedicated teacher. The school was also famous because of its quite unusual by Soviet standards sys- tem of self-government by the students. Quite a few graduates of the school later became well-known scientists, and really creative researchers. In 1966 Nikolai became a student at the Department of Mathematics and Mechanics of Odessa State University. Already at the third year of studies, he began his serious mathematical work under the supervision of the well-known Soviet mathematician Georgiy Semenovich Litvinchuk. Litvinchuk was a gifted teacher and scientific adviser. He, as anyone else, was capable of fascinating his students by new problems which have been always interesting and up-to-date. The weekly Odessa seminar on boundary value problems, chaired by Prof. Litvinchuk for more than 25 years, very much influenced Nikolai Vasilevski as well as others students of G.S. Litvinchuk. N. Vasilevski started to work on the problem of developing the Fredholm the- ory for a class of integral operators with nonintegrable integral kernels. In essence, the integral kernel was the Cauchy kernel multiplied by a logarithmic factor. The integral operators of this type lie between the singular integral operators and the integral operators whose kernels have weak (integrable) singularities. A famous Soviet mathematician F.D. Gakhov posted this problem in early 1950ies, and it remained open for more than 20 years. Nikolai managed to provide a complete so- lution in the setting which was much more general than the original. Working on
- 13. 2 S. Grudsky, Y. Latushkin and M. Shapiro this problem, Nikolai has demonstrated one of the main traits of his mathematical talent: his ability to achieve a deep penetration in the core of the problem, and to see rather unexpected connections between different theories. For instance, in order to solve Gakhov’s Problem, Nikolai utilized the theory of singular integral operators with coefficients having discontinuities of first kind, and the theory of operators whose integral kernels have fixed singularities – both theories just ap- peared at that time. The success of the young mathematician was well recognized by a broad circle of experts working in the area of boundary value problems and operator theory. In 1971 Nikolai was awarded the prestigious M. Ostrovskii Prize, given to the young Ukrainian scientists for the best research work. Due to his solu- tion of the famous problem, Nikolai quickly entered the mathematical community, and became known to many prominent mathematicians of that time. In particular, he was very much influenced by the his regular interactions with such outstanding mathematicians as M.G. Krein and S.G. Mikhlin. In 1973 N. Vasilevski defended his PhD thesis entitled “To the Noether theory of a class of integral operators with polar-logarithmic kernels”. In the same year he became an Assistant Professor at the Department of Mathematica and Mechanics of Odessa State University, where he was later promoted to the rank of Associate Professor, and, in 1989, to the rank of Full Professor. Having received the degree, Nikolai continued his active mathematical work. Soon, he displayed yet another side of his talent in approaching mathematical problems: his vision and ability to use general algebraic structures in operator theory, which, on one side, simplify the problem, and, on another, can be used in many other problems. We will briefly describe two examples of this. The first example is the method of orthogonal projections. In 1979, study- ing the algebra of operators generated by the Bergman projection, and by the operators of multiplication by piece-wise continuous functions, N. Vasilevski gave a description of the C∗ -algebra generated by two self-adjoint elements s and n satisfying the properties s2 + n2 = e and sn + ns = 0. A simple substitution p = (e + s − n)/2 and q = (e − s − n)/2 shows that this algebra is also generated by two self-adjoint idempotents (orthogonal projections) p and q (and the identity element e). During the last quarter of the past century, the latter algebra has been rediscovered by many authors all over the world. Among all algebras generated by orthogonal projections, the algebra generated by two projections is the only tame algebra (excluding the trivial case of the algebra with identity generated by one orthogonal projection). All algebras generated by three or more orthogonal projections are known to be wild, even when the projections satisfy some addi- tional constrains. Many model algebras arising in operator theory are generated by orthogonal projections, and thus any information of their structure essentially broadens the set of operator algebras admitting a reasonable description. In par- ticular, two and more orthogonal projections naturally appear in the study of various algebras generated by the Bergman projection and by piece-wise contin- uous functions having two or more different limiting values at a point. Although these projections, say, P, Q1, . . . , Qn, satisfy an extra condition Q1 +· · ·+Qn = I,
- 14. The Life and Work of Nikolai Vasilevski 3 they still generate, in general, a wild C∗ -algebra. At the same time, it was shown that the structure of the algebra just mentioned is determined by the joint prop- erties of certain positive injective contractions Ck, k = 1, . . . , n, satisfying the identity n k=1 Ck = I, and, therefore, the structure is determined by the structure of the C∗ -algebra generated by the contractions. The principal difference between the case of two projections and the general case of a finite set of projections is now completely clear: for n = 2 (and the projections P and Q + (I − Q) = I) we have only one contraction, and the spectral theorem directly leads to the de- sired description of the algebra. For n ≥ 2 we have to deal with the C∗ -algebra generated by a finite set of noncommuting positive injective contractions, which is a wild problem. Fortunately, for many important cases related to concrete oper- ator algebras, these projections have yet another special property: the operators PQ1P, . . . , PQnP mutually commute. This property makes the respective algebra tame, and thus it has a nice and simple description as the algebra of all n × n matrix-valued functions that are continuous on the joint spectrum Δ of the oper- ators PQ1P, . . . , PQnP, and have certain degeneration on the boundary of Δ. Another notable example of the algebraic structures used and developed by N. Vasilevski is his version of the Local Principle. The notion of locally equiva- lent operators, and localization theory were introduced and developed by I. Si- monenko in mid-sixtieth. According to the tradition of that time, the theory was focused on the study of individual operators, and on the reduction of the Fred- holm properties of an operator to local invertibility. Later, different versions of the local principle have been elaborated by many authors, including, among others, G.R. Allan, R. Douglas, I.Ts. Gohberg and N.Ia. Krupnik, A. Kozak, B. Silber- mann. In spite of the fact that many of these versions are formulated in terms of Banach- or C∗ -algebras, the main result, as before, reduces invertibility (or the Fredholm property) to local invertibility. On the other hand, at about the same time, several papers on the description of algebras and rings in terms of continuous sections were published by J. Dauns and K.H. Hofmann, M.J. Dupré, J.M.G. Fell, M. Takesaki and J. Tomiyama. These two directions have been developed inde- pendently, with no known links between the two series of papers. N. Vasilevski was the one who proposed a local principle which gives the global description of the algebra under study in terms of continuous sections of a certain canonically defined C∗ -bundle. This approach is based on general constructions of J. Dauns and K.H. Hofmann, and results of J. Varela. The main contribution consists of a deep re-comprehension of the traditional approach to the local principles uni- fying the ideas coming from both directions mentioned above, which results in a canonical procedure that provides the global description of the algebra under con- sideration in terms of continuous sections of a C∗ -bundle constructed by means of local algebras. In the eighties and even later, the main direction of the work of Nikolai Vasilevski has been the study of multi-dimensional singular integral operators with discontinuous coefficients. The main philosophy here was to study first algebras
- 15. 4 S. Grudsky, Y. Latushkin and M. Shapiro containing these operators, thus providing a solid foundation for the study of var- ious properties (in particular, the Fredholm property) of concrete operators. The main tool has been the described above version of the local principle. This princi- ple was not merely used to reduce the Fredholm property to local invertibility but also for a global description of the algebra as a whole based on the description of the local algebras. Using this methodology, Nikolai Vasilevski obtained deep re- sults in the theory of operators with Bergman’s kernel and piece-wise continuous coefficients, in the theory of multi-dimensional Toeplitz operators with pseudo- differential presymbols, in the theory of multi-dimensional Bitsadze operators, in the theory of multi-dimensional operators with shift, etc. In 1988 N. Vasilevski defended the Doctor of Sciences dissertation, based on these results, and entitled “Multi-dimensional singular integral operators with discontinuous classical sym- bols”. Besides being a very active mathematician, N. Vasilevski has been an excel- lent lecturer. His lectures are always clear, and sparkling, and full of humor, which so natural for someone who grew up in Odessa, a city with a longstanding tradi- tion of humor and fun. He was the first at Odessa State University who designed and started to teach a class in general topology. Students happily attended his lectures in Calculus, Real Analysis, Complex Analysis, Functional Analysis. He has been one of the most popular professor at the Department of Mathematics and Mechanics of Odessa State University. Nikolai is a master of presentations, and his colleagues always enjoy his talks at conferences and seminars. In 1992 Nikolai Vasilevski moved to Mexico. He started his career there as an Investigator (Full Professor) at the Mathematics Department of CINVESTAV (Centro de Investagacion y de Estudios Avansados). His appointment significantly strengthen the department which is one of the leading mathematical centers in Mexico. His relocation also visibly revitalized mathematical activity in the country in the field of operator theory. Actively pursuing his own research agenda, Nikolai also served as the organizer of several important conferences. For instance, let us mention the (regular since 1998) annual workshop “Análisis Norte-Sur”, and the well-known international conference IWOTA-2009. He initiated the relocation to Mexico a number of active experts in operator theory such as Yu. Karlovich and S. Grudsky, among others. During his tenure in Mexico, Nikolai Vasilevski produced a sizable group of students and younger colleagues; five of young mathematicians received PhD under his supervision. The contribution of N. Vasilevski in the theory of multi-dimensional singular integral operators found its rather unexpected development in his work on quater- nionic and Clifford analysis, published mainly with M. Shapiro in 1985–1995, start- ing still in the Soviet Union, with the subsequent continuation during the Mexican period of his life. Among others, the following topics have been considered: The settings for the Riemann boundary value problem for quaternionic functions that are taking into account both the noncommutative nature of quaternionic multi- plication and the presence of a family of classes of hyperholomorphic functions,
- 16. The Life and Work of Nikolai Vasilevski 5 which adequately generalize the notion of holomorphic functions of one complex variable; algebras, generated by the singular integral operators with quaternionic Cauchy kernel and piece-wise continuous coefficients; operators with quaternion and Clifford Bergman kernels. The Toeplitz operators in quaternion and Clifford setting have been introduced and studied in the first time. This work found the most favorable response and initiated dozens of citations. During his life in Mexico, the scientific interests of Nikolai Vasilevski mainly concentrated around the theory of Toeplitz operators on Bergman and Fock spaces. In the end of 1990ies, N. Vasilevski discovered a quite surprising phenomenon in the theory of Toeplitz operators on the Bergman space. Unexpectedly, there exists a rich family of commutative C∗ -algebras generated by Toeplitz operators with non-trivial defining symbols. In 1995 B. Korenblum and K. Zhu proved that the Toeplitz operators with radial defining symbols acting on the Bergman space over the unit disk can be diagonalized with respect to the standard monomial basis in the Bergman space. The C∗ -algebra generated by such Toeplitz operators is therefore obviously commutative. Four years later N. Vasilevski also showed the commutativity of the C∗ -algebra generated by the Toeplitz operators acting on the Bergman space over the upper half-plane and with defining symbols depend- ing only on Im z. Furthermore, he discovered the existence of a rich family of commutative C∗ -algebras of Toeplitz operators. Moreover, it turned out that the smoothness properties of the symbols do not play any role in commutativity: the symbols can be merely measurable. Surprisingly, everything is governed by the geometry of the underlying manifold, the unit disk equipped with the hyperbolic metric. The precise description of this phenomenon is as follows. Each pencil of hyperbolic geodesics determines the set of symbols which are constant on the cor- responding cycles, the orthogonal trajectories to geodesics forming the pencil. The C∗ -algebra generated by the Toeplitz operators with such defining symbols is com- mutative. An important feature of such algebras is that they remain commutative for the Toeplitz operators acting on each of the commonly considered weighted Bergman spaces. Moreover, assuming some natural conditions on “richness” of the classes of symbols, the following complete characterization has been obtained: A C∗ -algebra generated by the Toeplitz operators is commutative on each weighted Bergman space if and only if the corresponding defining symbols are constant on cycles of some pencil of hyperbolic geodesics. Apart from its own beauty, this result reveals an extremely deep influence of the geometry of the underlying manifold on the properties of the Toeplitz operators over the manifold. In each of the mentioned above cases, when the algebra is commutative, a certain unitary operator has been constructed. It reduces the corresponding Toeplitz operators to certain multiplica- tion operators, which also allows one to describe their representations of spectral type. This gives a powerful research tool for the subject, in particular, yielding direct access to the majority of the important properties such as boundedness, compactness, spectral properties, invariant subspaces, of the Toeplitz operators under study.
- 17. 6 S. Grudsky, Y. Latushkin and M. Shapiro The results of the research in this directions became a part of the monograph “Commutative Algebras of Toeplitz Operators on the Bergman Space” published by N. Vasilevski in Birkhäuser in 2008. Nikolai Leonidovich Vasilevski passed his sixties birthday on full speed, and being in excellent shape. We, his friends, students, and colleagues, wish him further success and, above all, many new interesting and successfully solved problems. Principal publications of Nikolai Vasilevski Book 1. N.L. Vasilevski. Commutative Algebras of Toeplitz Operators on the Bergman Space, Operator Theory: Advances and Applications, Vol. 183, Birkhäuser Verlag, Basel-Boston-Berlin, 2008, XXIX, 417 p. Articles 1. N.L. Vasilevski. On the Noether conditions and a formula for the index of a class of integral operators. Doklady Akad. Nauk SSSR, 1972, v. 202, No 4, p. 747–750 (Russian). English translation: Soviet Math. Dokl., v. 13, no. 1, 1972, p. 175–179. 2. N.L. Vasilevski. On properties of a class of integral operators in the space Lp. Matemat. Zametki, 1974, v. 16, No 4, p. 529–535 (Russian). English translation: Math. Notes, v. 16, no. 4, 1974, p. 905–909. 3. N.L. Vasilevski. The Noether theory of a class of potential type integral oper- ators. Izvestija VUZov. Matematika, 1974, No 7, p. 12–20 (Russian). English translation: Soviet Math. (Izv. VUZ), v. 18, no. 7, 1974, p. 8–15. 4. N.L. Vasilevski. On the Noetherian theory of integral operators with a polar logarithmic kernel. Doklady Akad. Nauk SSSR, 1974, v. 215, No 3, p. 514–517 (Russian). English translation: Soviet Math. Dokl., v. 15, no. 2, 1974, p. 522–527. 5. N.L. Vasilevski, E.V. Gutnikov. On the symbol of operators forming finite- dimensional algebras. Doklady Akad. Nauk SSSR, 1975, v. 221, No 1, p. 18–21 (Russian). English translation: Soviet Math. Dokl., v. 16, no. 2, 1975, p. 271–275. 6. N.L. Vasilevski, G.S. Litvinchuk. Theory of solvability of a class of singular integral equations with involution. Doklady Akad. Nauk SSSR, 1975, v. 221, No 2, p. 269–271 (Russian). English translation: Soviet Math. Dokl., v. 16, no. 2, 1975, p. 318–321. 7. N.L. Vasilevski, M.V. Shapiro. On an algebra generated by singular integral operators with the Carleman shift and in the case of piece-wise continuous coefficients. Ukrainski Matematicheski Zurnal, 1975, v. 27, No 2, p. 216–223 (Russian). English translation: Ukrainian Math. J., v. 27, no. 2, 1975, p. 171–176.
- 18. The Life and Work of Nikolai Vasilevski 7 8. N.L. Vasilevski. On a class of singular integral operators with kernels of polar- logarithmic type. Izvestija Akad. Nauk SSSR, ser. matem., 1976, v. 40, No 1, p. 131–151 (Russian). English translation: Math. USSR Izvestija, v. 10, no. 1, 1976, p. 127–143. 9. N.L. Vasilevski, E.V. Gutnikov. On the structure of the symbol of operators forming finite-dimensional algebras. Doklady Akad. Nauk SSSR, 1976, v. 230, No 1, p. 11–14 (Russian). English translation: Soviet Math. Dokl., v. 17, no. 5, 1976, p. 1225–1229. 10. N.L. Vasilevski, A.A. Karelin, P.V. Kerekesha, G.S. Litvinchuk. On a class of singular integral equations with shift and its applications in the theory of boundary value problems for partial differential equations. I. Differentsialnye Uravnenija, 1977, v. 13, No 9, p. 1692–1700 (Russian). English translation: Diff. Equations, v. 13, no. 9, 1977, p. 1180–1185. 11. N.L. Vasilevski, A.A. Karelin, P.V. Kerekesha, G.S. Litvinchuk. On a class of singular integral equations with shift and its applications in the theory of boundary value problems for partial differential equations. II. Differentsialnye Uravnenija, 1977, v. 13, No 11, p. 2051–2062 (Russian). English translation: Diff. Equations, v. 13, no. 11, 1977, p. 1430–1438. 12. N.L. Vasilevski. Symbols of operator algebras. Doklady Akad. Nauk SSSR, 1977, v. 235, No 1, p. 15–18 (Russian). English translation: Soviet Math. Dokl., v. 18, no. 4, 1977, p. 872–876. 13. N.L. Vasilevski, A.A. Karelin. An investigation of a boundary value problem for the partial differential equation of the mixed type with the help of reduc- tion to the singular integral equation with Carleman shift. Izvestija VUZov. Matematika, 1978, No 3, p. 15–19, (Russian). English translation: Soviet Math. (Izv. VUZ), v. 22, no. 3, 1978, p. 11–15. 14. N.L. Vasilevski, R. Trujillo. On ΦR-operators in matrix algebras of operators. Doklady Akad. Nauk SSSR, 1979, v. 245, No 6, p. 1289–1292 (Russian). English translation: Soviet Math. Dokl., v. 20, no. 2, 1979, p. 406–409. 15. N.L. Vasilevski, R. Trujillo. On the theory of ΦR-operators in matrix algebras of operators. Linejnye Operatory, Kishinev, 1980, p. 3–15 (Russian). 16. N.L. Vasilevski. On an algebra generated by some two-dimensional integral operators with continuous coefficients in a subdomain of the unit disc. Journal of Integral Equations, 1980, v. 2, p. 111–116. 17. N.L. Vasilevski. Banach algebras generated by some two-dimensional integral operators. I. Math. Nachr., 1980, b. 96, p. 245–255 (Russian). 18. N.L. Vasilevski. Banach algebras generated by some two-dimensional integral operators. II. Math. Nachr., 1980, b. 99, p. 136–144 (Russian). 19. N.L. Vasilevski. On the symbol theory for Banach operator algebras which generalizes algebras of singular integral operators. Differentsialnye Uravnen- nija, 1981, v. 17, No 4, p. 678–688 (Russian). English translation: Diff. Equations, v. 17, no. 4, 1981, p. 462–469.
- 19. 8 S. Grudsky, Y. Latushkin and M. Shapiro 20. N.L. Vasilevski, I.M. Spitkovsky. On an algebra generated by two projectors. Doklady Akad. Nauk UkSSR, Ser. “A”, 1981, No 8, p. 10–13 (Russian). 21. N.L. Vasilevski. On the algebra generated by two-dimensional integral oper- ators with Bergman kernel and piece-wise continuous coefficients. Doklady Akad. Nauk SSSR, 1983, v. 271, No 5, p. 1041–1044 (Russian). English translation: Soviet Math. Dokl., v. 28, no. 1, 1983, p. 191–194. 22. N.L. Vasilevski. On certain algebras generated by a space analog of the sin- gular operator with Cauchy kernel. Doklady Akad. Nauk SSSR, 1983, v. 273, No 3, p. 521–524 (Russian). English translation: Soviet Math. Dokl., v. 28, no. 3, 1983, p. 654–657. 23. N.L. Vasilevski. On an algebra generated by abstract singular operators and Carleman shift. Soobshchenija Akad. Nauk GSSR, 1984, v. 115, No 3, p. 473–476 (Russian). 24. N.L. Vasilevski. On an algebra generated by multivariable Wiener-Hopf oper- ators. Reports of Enlarged Session of Seminars of the I.N. Vekua Institute of Applied Mathematics. Tbilisi, 1985, v. 1, p. 59–62 (Russian). 25. N.L. Vasilevski, M.V. Shapiro. On an analogy of monogenity in the sense of Moisil-Teodoresko and some applications in the theory of boundary value problems. Reports of Enlarged Sessions of Seminars of the I.N. Vekua Institute of Applied Mathematics. Tbilisi, 1985, v. 1, p. 63–66 (Russian). 26. N.L. Vasilevski. Algebras generated by multivariable Toeplitz operators with piece-wise continuous presymbols. Scientific Proceedings of the Boundary Value Problems Seminar Dedicated to 75th birthday of Academician BSSR Academy of Sciences F.D. Gahov. Minsk, 1985, p. 149–150 (Russian). 27. N.L. Vasilevski. Two-dimensional Mikhlin-Calderon-Zygmund operators and bisingular operators. Sibirski Matematicheski Zurnal, 1986, v. 27, No 2, p. 23–31 (Russian). English translation: Siberian Math. J., v. 27, no. 2, 1986, p. 161–168. 28. N.L. Vasilevski. Banach algebras generated by two-dimensional integral oper- ators with Bergman Kernel and piece-wise continuous coefficients. I. Izvestija VUZov, Matematika, 1986, No 2, p. 12–21 (Russian). English translation: Soviet Math. (Izv. VUZ), v. 30, no. 2, 1986, p. 14–24. 29. N.L. Vasilevski. Banach algebras generated by two-dimensional integral opera- tors with Bergman Kernel and piece-wise continuous coefficients. II. Izvestija VUZov, Matematika, 1986, No 3, p. 33–38 (Russian). English translation: Soviet Math. (Izv. VUZ), v. 30, no. 3, 1986, p. 44–50. 30. N.L. Vasilevski. Algebras generated by multidimensional singular integral op- erators and by coefficients admitting discontinuities of homogeneous type. Matematicheski Sbornik, 1986, v. 129, No 1, p. 3–19 (Russian). English translation: Math. USSR Sbornik, v. 57, no. 1, 1987, p. 1–19.
- 20. The Life and Work of Nikolai Vasilevski 9 31. N.L. Vasilevski. On an algebra generated by Toeplitz operators with zero-order pseudodifferential presymbols. Doklady Akad. Nauk SSSR, 1986, v. 289, No 1, p. 14–18 (Russian). English translation: Soviet Math. Dokl., v. 34, no. 1, 1987, p. 4–7. 32. N.L. Vasilevski, M.V. Shapiro. On quaternion Ψ-monogenic function. “Meth- ods of solving of the direct and inverse geoelectrical problems”. 1987, p. 54–65 (Russian). 33. N.L. Vasilevski. On an algebra connected with Toeplitz operators on the tube domains. Izvestija Akad. Nauk SSSR, ser. matem., 1987, v. 51, No 1, p. 79–95 (Russian). English translation: Math. USSR Izvestija, v. 30, no.1, 1988, p. 71–87. 34. N.L. Vasilevski, R. Trujillo. On C∗ -algebra generated by almost-periodic two- dimensional singular integral operators with discontinuous presymbols. Funk- cionalny Analiz i ego Prilogenija, 1987, v. 21, No 3, p. 75–76 (Russian). English translation: Func. Analysis and its Appl., v. 21, no. 3, 1987, p. 235– 236. 35. N.L. Vasilevski. Toeplitz operators associated with the Siegel domains. Matem- aticki Vesnik, 1988, v. 40, p. 349–354. 36. N.L. Vasilevski. Hardy spaces associated with the Siegel domains. Reports of Enlarged Sessions of Seminars of the I.N. Vekua Institute of Applied Math- ematics. Tbilisi, 1988, v. 3, No 1, p. 48–51 (Russian). 37. N.L. Vasilevski, M.V. Shapiro. Holomorphy, hyperholomorphy Toeplitz oper- ators. Uspehi Matematicheskih Nauk, 1989, v. 44, No 4 (268), p. 226–227 (Russian). English translation: Russian Math. Surveys, v. 44, no. 4, 1989, p. 196–197. 38. N.L. Vasilevski, M.V. Shapiro. Some questions of hypercomplex analysis “Complex Analysis and Applications ’87”, Sofia, 1989, p. 523–531. 39. N.L. Vasilevski. Non-classical singular integral operators and algebras gen- erated by them. Integral Equations and Boundary Value Problems. World Scientific. 1991, p. 210–215. 40. M.V. Shapiro, N.L. Vasilevski. Singular integral operator in Clifford analysis, Clifford Algebras and Their Applications in Mathematical Physics, Kluwer Academic Publishers, Netherlands, 1992, p. 271–277. 41. N.L. Vasilevski. On an algebra generated by abstract singular operators and a shift operator. Math. Nachr., v. 162, 1993, p. 89–108. 42. R.M. Porter, M.V. Shapiro, N.L. Vasilevski. On the analogue of the ∂-problem in quaternionic analysis. Clifford Algebras and Their Applications in Mathe- matical Physics, F. Brackx et al., eds., Kluwer Academic Publishers, Nether- lands, 1993, p. 167–173.
- 21. 10 S. Grudsky, Y. Latushkin and M. Shapiro 43. E. Ramı́rez de Arellano, M.V. Shapiro, N.L. Vasilevski. Hurwitz pairs and Clifford algebra representations. Clifford Algebras and Their Applications in Mathematical Physics, F. Brackx et al., eds., Kluwer Academic Publishers, Netherlands, 1993, p. 175–181. 44. M.V. Shapiro, N.L. Vasilevski. On the Bergman kernel function in the Clifford analysis. Clifford Algebras and Their Applications in Mathematical Physics, F. Brackx et al., eds., Kluwer Academic Publishers, Netherlands, 1993, p. 183–192. 45. N.L. Vasilevski. On “discontinuous” boundary value problems for pseudodif- ferential operators. International Conference on Differential Equations, Vol. 1, 2, (Barcelona, 1991), World Sci. Publishing, River Edge, NJ, 1993, p. 953– 958. 46. N.L. Vasilevski, R. Trujillo. Convolution operators on standard CR-manifolds. I. Structural Properties. Integral Equations and Operator Theory, v. 19, no. 1, 1994, p. 65–107. 47. N.L. Vasilevski. Convolution operators on standard CR-manifolds. II. Alge- bras of convolution operators on the Heisenberg group. Integral Equations and Operator Theory, v. 19, no. 3, 1994, p. 327–348. 48. N.L. Vasilevski. On an algebra generated by two-dimensional singular integral operators in plane domains. Complex Variables, v. 26, 1994, p. 79–91. 49. R.M. Porter, M. Shapiro, N. Vasilevski. Quaternionic differential and integral operators and the ∂-problem. Journal of Natural Geometry, v. 6, no. 2, 1994, p. 101–124. 50. E. Ramı́rez de Arellano, M.V. Shapiro, N.L. Vasilevski. Two types of analysis associated to the notion of Hurwitz pairs. Differential Geometric Methods in Theoretical Physics, Ed. J. Keller, Z. Oziewich, Advances in Applied Clifford Algebras, v. 4 (S1), 1994, p. 413–422. 51. M.V. Shapiro, N.L. Vasilevski. Quaternionic Ψ-hyperholomorphic functions, singular integral operators and boundary value problems. I. Ψ-hyperholo- morphic function theory. Complex Variables, v. 27, 1995, p. 17–46. 52. M.V. Shapiro, N.L. Vasilevski. Quaternionic Ψ-hyperholomorphic functions, singular integral operators and boundary value problems. II. Algebras of sin- gular integral operators and Riemann type boundary value problems. Complex Variables, v. 27, 1995, p. 67–96. 53. N. Vasilevski, V. Kisil, E. Ramirez de Arellano, R. Trujillo. Toeplitz operators with discontinuous presymbols on the Fock space. Russian Math. Doklady, v. 345, no. 2, 1995, p. 153–155 (Russian). English translation: Russian Math. Doklady. 54. E. Ramı́rez de Arellano, N.L. Vasilevski. Toeplitz operators on the Fock space with presymbols discontinuous on a thick set, Mathematische Nachrichten, v. 180, 1996, p. 299–315.
- 22. The Life and Work of Nikolai Vasilevski 11 55. E. Ramı́rez de Arellano, N.L. Vasilevski. Algebras of singular integral opera- tors generated by three orthogonal projections, Integral Equations and Oper- ator Theory, v. 25, no. 3, 1996, p. 277–288. 56. N. Vasilevski, E. Ramirez de Arellano, M. Shapiro. Hurwitz classical problem and associated function theory. Russian Math. Doklady, v. 349, no. 5, 1996, p. 588–591 (Russian). English translation: Russian Math. Doklady 57. E. Ramı́rez de Arellano, M.V. Shapiro, N.L. Vasilevski. The hyperholomor- phic Bergman projector and its properties. In: Clifford Algebras and Related Topics, J. Ryan, Ed. CRC Press, Chapter 19, 1996, p. 333–344. 58. E. Ramı́rez de Arellano, M.V. Shapiro, N.L. Vasilevski. Hurwitz analysis: basic concepts and connection with Clifford analysis. In: Generalizations of Complex Analysis and their Applications in Physics, J. Lawrynowicz, Ed. Banach Center Publications, V. 37, Warszawa, 1996, p. 209–221. 59. M.V. Shapiro, N.L. Vasilevski. On the Bergman kernel function in hyperholo- morphic analysis, Acta Applicandae Mathematicae, v. 46, 1997, p. 1–27. 60. E. Ramı́rez de Arellano, N.L. Vasilevski. Bargmann projection, three-valued functions and corresponding Toeplitz operators, Contemporary Mathematics, v. 212, 1998, p. 185–196. 61. N.L. Vasilevski. C*-algebras generated by orthogonal projections and their applications. Integral Equations and Operator Theory, v. 31, 1998, p. 113– 132. 62. N.L. Vasilevski, M.V. Shapiro. On the Bergman kern-function on quater- nionic analysis. Izvestiia VUZov, Matematika, no. 2, 1998, p. 84–88 (Rus- sian). English translation: Russian Math. (Izvestiia VUZ), v. 42, no. 2, 1998, p. 81–85. 63. N.L. Vasilevski, On the structure of Bergman and poly-Bergman spaces, In- tegral Equations and Operator Theory, v. 33, 1999, p. 471–488. 64. N.L. Vasilevski, On Bergman-Toeplitz operators with commutative symbol al- gebras, Integral Equations and Operator Theory, v. 34, no. 1, 1999, p. 107– 126. 65. E. Ramı́rez de Arellano, M.V. Shapiro, N.L. Vasilevski. The hyperholomor- phic Bergman projector and its properties. In: Clifford Algebras and Related Topics, J. Ryan, Ed. CRC Press, Chapter 19, 1996, p. 333–344. 66. E. Ramı́rez de Arellano, M.V. Shapiro, N.L. Vasilevski. Hurwitz analysis: basic concepts and connection with Clifford analysis. In: Generalizations of Complex Analysis and their Applications in Physics, J. Lawrynowicz, Ed. Banach Center Publications, v. 37, Warszawa, 1996, p. 209–221. 67. N.L. Vasilevski, On quaternionic Bergman and poly-Bergman spaces, Com- plex Variables, v. 41, 2000, p. 111–132.
- 23. 12 S. Grudsky, Y. Latushkin and M. Shapiro 68. V.S. Rabinovish, N.L. Vasilevski, Bergman-Toeplitz and pseudodifferential operators, Operator Theory. Advances and Applications v. 114, 2000, p. 207– 234. 69. N.L. Vasilevski, Poly-Fock Spaces, Operator Theory. Advances and Applica- tions v. 117, 2000, p. 371–386. 70. V.V. Kucherenko, N.L. Vasilevski, A shift operator generated by a trigono- metric system, Mat. Zametki, v. 67, no. 4, 2000, p. 539–548 (Russian). English translation: Mat. Notes. 71. N.L. Vasilevski, The Bergman space in tube domains, and commuting Toeplitz operators, Doklady RAN, v. 372, no. 1, 2000, p. 9–12 (Russian). English translation: Doklady Mathematics, v. 61, no. 3, 2000. 72. N.L. Vasilevski. Bergman space on tube domains and commuting Toeplitz operators. In: Proceedings of the Second ISAAC Congress, Volume 2, H.G.W. Begehr et al. (eds.), Kluwer Academic Publishers, The Netherlands, Chapter 163, 2000, p. 1523–1537. 73. S. Grudsky, N. Vasilevski, Bergman-Toeplitz operators: Radial component in- fluence, Integral Equations and Operator Theory, v. 40, no. 1, 2001, p. 16–33. 74. A.N. Karapetyants, V.S. Rabinovich, N.L. Vasilevski, On algebras of two- dimensional singular integral operators with homogeneous discontinuities in symbols, Integral Equations and Operator Theory, v. 40, no. 3, 2001, p. 278– 308. 75. N.L. Vasilevski. Toeplitz Operators on the Bergman Spaces: Inside-the- Domain Effects, Contemporary Mathematics, v. 289, 2001, p. 79–146. 76. N.L. Vasilevski. Bergman spaces on the unit disk. In: Clifford Analysis and Its Applications F. Brackx et al. (eds.), Kluwer Academic Publishers, The Netherlands, 2001, p. 399–409. 77. S. Grudsky, N. Vasilevski. Toeplitz operators on the Fock space: Radial com- ponent effects, Integral Equations and Operator Theory, v. 44, no. 1, 2002, p. 10–37. 78. N.L. Vasilevski. Commutative algebras of Toeplitz operators and hyperbolic geometry. In: Proceedings of the Ukranian Mathematical Congress – 2001, Functional Analysis, Section 11, Institute of Mathematics of the National Academy of Sciences, Ukraine, 2002, p. 22–35. 79. N.L. Vasilevski. Bergman Space Structure, Commutative Algebras of Toeplitz Operators and Hyperbolic Geometry, Integral Equations and Operator The- ory, v. 46, 2003, p. 235–251. 80. S. Grudsky, A. Karapetyants, N. Vasilevski. Toeplitz Operators on the Unit Ball in Cn with Radial Symbols, J. Operator Theory, v. 49, 2003, p. 325–346. 81. J. Ramı́rez Ortega, N. Vasilevski, E. Ramı́rez de Arellano On the algebra gen- erated by the Bergman projection and a shift operator. I. Integral Equations and Operator Theory, v. 46, no. 4, 2003, p. 455–471.
- 24. The Life and Work of Nikolai Vasilevski 13 82. N.L. Vasilevski. Toeplitz operators on the Bergman space. In: Factorization, Singular Operators and Related Problems, Edited by S. Samko, A. Lebre, A.F. dos Santos, Kluwer Academic Publishers, 2003, p. 315–333. 83. J. Ramı́rez Ortega, E. Ramı́rez de Arellano, N. Vasilevski On the algebra generated by the Bergman projection and a shift operator. II. Bol. Soc. Mat. Mexicana (3), v. 10, 2004, p. 105–117. 84. S. Grudsky, A. Karapetyants, N. Vasilevski. Dynamics of properties of Toeplitz operators on the upper half-plane: Hyperbolic case, Bol. Soc. Mat. Mexicana (3), v. 10, 2004, p. 119–138. 85. S. Grudsky, A. Karapetyants, N. Vasilevski. Dynamics of properties of Toeplitz operators on the upper half-plane: Parabolic case, J. Operator The- ory, v. 52, no. 1, 2004, p. 185–204. 86. S. Grudsky, A. Karapetyants, N. Vasilevski. Dynamics of properties of Toeplitz operators with radial Symbols, Integral Equations and Operator The- ory, v. 50, no. 2, 2004, p. 217–253. 87. N.L. Vasilevski. On a general local principle for C∗ -algebras, Izv.VUZ North- Caucasian Region, Natural Sciences, Special Issue, “Pseudodifferential oper- ators and some problems of mathematical physics”, 2005, p. 34–42 (Russian). 88. S. Grudsky, N. Vasilevski. Dynamics of Spectra of Toeplitz Operators, Ad- vances in Analysis. Proceedings of the 4th International ISAAC Congress. (York University, Toronto, Canada 11–16 August 2003), World Scientific, New Jersey London Singapore, 2005, p. 495–504. 89. S. Grudsky, R. Quiroga-Barranco, N. Vasilevski. Commutative C∗ -algebras of Toeplitz operators and quantization on the unit disk, J. Functional Analysis, v. 234, 2006, p. 1–44. 90. N.L. Vasilevski, S.M. Grudsky, A.N. Karapetyants. Dynamics of properties of Toeplitz operators on weighted Bergman spaces, Siberian Electronic Math. Reports, v. 3, 2006, p. 362–383 (Russian). 91. N. Vasilevski. On the Toeplitz operators with piecewise continuous symbols on the Bergman space, In: “Modern Operator Theory and Applications”, Operator Theory: Advances and Applications, v. 170, 2007, p. 229–248. 92. N. Vasilevski. Poly-Bergman spaces and two-dimensional singular integral operators, Operator Theory: Advances and Applications, v. 171, 2007, p. 349–359. 93. N. Tarkhanov, N. Vasilevski. Microlocal analysis of the Bochner-Martinelli integral, Integral Equations and Operator Theory, v. 57, 2007, p. 583–592. 94. R. Quiroga-Barranco, N. Vasilevski. Commutative algebras of Toeplitz opera- tors on the Reinhardt domains, Integral Equations and Operator Theory, v. 59, no. 1, 2007, p. 67–98.
- 25. 14 S. Grudsky, Y. Latushkin and M. Shapiro 95. R. Quiroga-Barranco, N. Vasilevski. Commutative C∗ -algebras of Toeplitz op- erators on the unit ball, I. Bargmann-type transforms and spectral represen- tations of Toeplitz operators, Integral Equations and Operator Theory, v. 59, no. 3, 2007, p. 379–419. 96. R. Quiroga-Barranco, N. Vasilevski. Commutative C∗ -algebras of Toeplitz op- erators on the unit ball, II. Geometry of the level sets of symbols, Integral Equations and Operator Theory, v. 60, no. 1, 2008, p. 89–132. 97. N. Vasilevski. Commutative algebras of Toeplitz operators and Berezin quan- tization, Contemporary Mathematics, v. 462, 2008, p. 125–143. 98. S. Grudsky, N. Vasilevski. On the structure of the C∗ -algebra generated by Toeplitz operators with piece-wise continuous symbols, Complex Analysis and Operator Theory, v. 2, no. 4, 2008, p. 525–548. Ph. D. dissertations directed by Nikolai Vasilevski 1. Rafael Trujillo, Fredholm Theory of Tensor Product of Operator Algebras, Odessa State University, 1986. 2. Zhelko Radulovich, Algebras of Multidimensional Singular Integral Opera- tors with Discontinuous Symbols with Respect to Dual Variable, Odessa State University, 1991. 3. Vladimir Kisil, Algebras of Pseudodifferential Operators Associated with the Heisenberg Group, Odessa State University, 1992. 4. Josué Ramı́rez Ortega, Algebra generada por la proyección de Bergman y un operador de translación, CINVESTAV del I.P.N., Mexico City, 1999. 5. Maribel Loaiza Leyva, Algebra generada por la proyección de Bergman y por los operadores de multiplicación por funciones continuas a trozos, CINVES- TAV del I.P.N., Mexico City, 2000. 6. Ernesto Prieto Sanabrı́a, Operadores de Toeplitz en la 2-esfera en los espacios de Bergman con peso, CINVESTAV del I.P.N., Mexico City, 2007. 7. Armando Sánchez Nungaray, Super operadores de Toeplitz en la dos-esfera, CINVESTAV del I.P.N., Mexico City, 2008. 8. Carlos Moreno Muñoz, Operadores de Toeplitz en el espacio de Bergman con peso: Caso parabólico, CINVESTAV del I.P.N., Mexico City, 2009.
- 26. Operator Theory: Advances and Applications, Vol. 210, 15–36 c 2010 Springer Basel AG On the Structure of the Eigenvectors of Large Hermitian Toeplitz Band Matrices Albrecht Böttcher, Sergei M. Grudsky and Egor A. Maksimenko For Nikolai Vasilevski on His 60th Birthday Abstract. The paper is devoted to the asymptotic behavior of the eigenvec- tors of banded Hermitian Toeplitz matrices as the dimension of the matrices increases to infinity. The main result, which is based on certain assumptions, describes the structure of the eigenvectors in terms of the Laurent polynomial that generates the matrices up to an error term that decays exponentially fast. This result is applicable to both extreme and inner eigenvectors. Mathematics Subject Classification (2000). Primary 47B35; Secondary 15A18, 41A25, 65F15. Keywords. Toeplitz matrix, eigenvector, asymptotic expansions. 1. Introduction and main results Given a function a in L1 on the complex unit circle T, we denote by a the th Fourier coefficient, a = 1 2π 2π 0 a(eix )e−ix dx ( ∈ Z), and by Tn(a) the n × n Toeplitz matrix (aj−k)n j,k=1. We assume that a is real- valued, in which case the matrices Tn(a) are all Hermitian. Let λ (n) 1 ≤ λ (n) 2 ≤ · · · ≤ λ(n) n be the eigenvalues of Tn(a) and let {v (n) 1 , v (n) 2 , . . . , v(n) n } This work was partially supported by CONACYT project 80503, Mexico.
- 27. 16 A. Böttcher, S.M. Grudsky and E.A. Maksimenko be an orthonormal basis of eigenvectors such that Tn(a)v (n) j = λ (n) j v (n) j . The present paper is dedicated to the asymptotic behavior of the eigenvectors v (n) j as n → ∞. To get an idea of the kind of results we will establish, consider the function a(eix ) = 2−2 cosx. The range a(T) is the segment [0, 4]. It is well known that the eigenvalues and eigenvectors of Tn(a) are given by λ (n) j = 2 − 2 cos πj n + 1 , x (n) j = 2 n + 1 sin mπj n + 1 n m=1 . (1.1) (We denote the eigenvectors in this reference case by x (n) j and reserve the notation v (n) j for the general case.) Let ϕ be the function ϕ : [0, 4] → [0, π], ϕ(λ) = arccos 2 − λ 2 . We have ϕ(λ (n) j ) = πj/(n + 1) and hence, apart from the normalization factor 2/(n + 1), x (n) j,m is the value of sin(mϕ(λ)) at λ = λ (n) j . In other words, an eigenvector for λ is given by (sin(mϕ(λ)))n m=1. A speculative question is whether in the general case we can also find functions Ωm such that, at least asymptotically, (Ωm(λ))n m=1 is an eigenvector for λ. It turns out that this is in general impossible but that after a slight modification the answer to the question is in the affirmative. Namely, we will prove that, under certain assumptions, there are functions Ωm, Φm and real-valued functions σ, η such that an eigenvector for λ = λ (n) j is always of the form Ωm(λ)+Φm(λ)+(−1)j+1 e−i(n+1)σ(λ) e−iη(λ) Φn+1−m(λ)+error term n m=1 . (1.2) The error term will be shown to decrease to zero exponentially fast and uniformly in j and m as n → ∞. Moreover, we will show that Ωm(λ) is an oscillating function of m for each fixed λ and that Φm(λ) decays exponentially fast to zero as m → ∞ for each λ (which means that Φn+1−m(λ) is an exponentially increasing function of m for each λ). Finally, it will turn out that n m=1 |Φm(λ)|2 n m=1 |Ωm(λ)|2 = O 1 n as n → ∞, uniformly in λ. Thus, the dominant term in (1.2) is Ωm(λ), while the terms containing Φm(λ) and Φn+1−m(λ) may be viewed as twin babies. If a is also an even function, a(eix ) = a(e−ix ) for all x, then all the matrices Tn(a) are real and symmetric. In [4], we conjectured that then, again under addi- tional but reasonable assumptions, the appropriately rotated extreme eigenvectors v (n) j are all close to the vectors x (n) j . To be more precise, we conjectured that if n → ∞ and j (or n − j) remains fixed, then there are complex numbers τ (n) j of
- 28. Eigenvectors of Large Hermitian Toeplitz Band Matrices 17 modulus 1 such that τ (n) j v (n) j − x (n) j 2 = o(1), (1.3) where · 2 is the 2 norm. Several results related to this conjecture were estab- lished in [3] and [4]. We here prove this conjecture under assumptions that will be specified in the following paragraph. We will even be able to show that the o(1) in (1.3) is O(j/n) if j/n → 0 and O(1 − j/n) if j/n → 1. Throughout what follows we assume that a is a Laurent polynomial a(t) = r k=−r aktk (t = eix ∈ T) with r ≥ 2, ar = 0, and ak = a−k for all k. The last condition means that a is real-valued on T. We assume without loss of generality that a(T) = [0, M] with M 0 and that a(1) = 0 and a(eiϕ0 ) = M for some ϕ0 ∈ (0, 2π). We require that the function g(x) := a(eix ) is strictly increasing on (0, ϕ0) and strictly decreasing on (ϕ0, 2π) and that the second derivatives of g at x = 0 and x = ϕ0 are nonzero. Finally, we denote by [α, β] ⊂ [0, M] a segment such that if λ ∈ [α, β], then the 2r − 2 zeros of the Laurent polynomial a(z) − λ that lie in C T are pairwise distinct. Note that we exclude the case r = 1, because in this case the eigenvalues and eigenvectors of Tn(a) are explicitly available. Also notice that if r = 2, which is the case of pentadiagonal matrices, then for every λ ∈ [0, M] the polynomial a(z) − λ has two zeros on T, one zero outside T, and one zero inside T. Thus, in this situation the last requirement of the previous paragraph is automatically satisfied for [α, β] = [0, M]. The asymptotic behavior of the extreme eigenvalues and eigenvectors of Tn(a), that is, of λ (n) j and v (n) j when j or n − j remain fixed, has been stud- ied by several authors. As for extreme eigenvalues, the pioneering works are [7], [9], [11], [12], [18], while recent papers on the subject include [3], [6], [8], [10], [13], [14], [15], [19], [20]. See also the books [1] and [5]. Much less is known about the asymptotics of the eigenvectors. Part of the results of [4] and [19] may be inter- preted as results on the behavior of the eigenvectors “in the mean” on the one hand and as insights into what happens if eigenvectors are replaced by pseudomodes on the other. In [3], we investigated the asymptotics of the extreme eigenvectors of certain Hermitian (and not necessarily banded) Toeplitz matrices. Our paper [2] may be considered as a first step to the understanding of the asymptotic behavior of individual inner eigenvalues of Toeplitz matrices. In the same vein, this paper intends to understand the nature of individual eigenvectors as part of the whole, independently of whether they are extreme or inner ones. To state our main results, we need some notation. Let λ ∈ [0, M]. Then there are uniquely defined ϕ1(λ) ∈ [0, ϕ0] and ϕ2(λ) ∈ [ϕ0 − 2π, 0] such that g(ϕ1(λ)) = g(ϕ2(λ)) = λ;
- 29. 18 A. Böttcher, S.M. Grudsky and E.A. Maksimenko recall that g(x) := a(eix ). We put ϕ(λ) = ϕ1(λ) − ϕ2(λ) 2 , σ(λ) = ϕ1(λ) + ϕ2(λ) 2 . We have a(z) − λ = z−r arz2r + · · · + (a0 − λ)zr + · · · + a−r = arz−r 2r k=1 (z − zk(λ)), and our assumptions imply that we can label the zeros zk(λ) so that the collection Z(λ) of the zeros may be written as {z1(λ), . . . , zr−1(λ), zr(λ), zr+1(λ), zr+2(λ), . . . , z2r(λ)} = {u1(λ), . . . , ur−1(λ), eiϕ1(λ) , eiϕ2(λ) , 1/u1(λ), . . . , 1/ur−1(λ)} (1.4) where |uν(λ)| 1 for 1 ≤ ν ≤ r − 1 and each uν(λ) depends continuously on λ ∈ [0, M]. Here and in similar places below we write uk(λ) := uk(λ). We define δ0 0 by eδ0 = min λ∈[0,M] min 1≤ν≤r−1 |uν(λ)|. Throughout the following, δ stands for any number in (0, δ0). Further, we denote by hλ the function hλ(z) = r−1 ν=1 1 − z uν(λ) . The function Θ(λ) = hλ(eiϕ1(λ) )/hλ(eiϕ2(λ) ) is continuous and nonzero on [0, M] and we have Θ(0) = Θ(M) = 1. In [2], it was shown that the closed curve [0, M] → C {0}, λ → Θ(λ) has winding number zero. Let θ(λ) be the continuous argument of Θ(λ) for which θ(0) = θ(M) = 0. In [2], we proved that if n is large enough, then the function fn : [0, M] → [0, (n + 1)π], fn(λ) = (n + 1)ϕ(λ) + θ(λ) is bijective and increasing and that if λ (n) j,∗ is the unique solution of the equation fn(λ (n) j,∗ ) = πj, then the eigenvalues λ (n) j satisfy |λj − λ (n) j,∗ | ≤ K e−δn for all j ∈ {1, . . . , n}, where K is a finite constant depending only on a. Thus, we have (n + 1)ϕ(λ (n) j ) + θ(λ (n) j ) = πj + O(e−δn ), (1.5) uniformly in j ∈ {1, . . ., n}.
- 30. Eigenvectors of Large Hermitian Toeplitz Band Matrices 19 Now take λ from (α, β). For j ∈ {1, . . ., n} and ν ∈ {1, . . . , r − 1}, we put A(λ) = eiσ(λ) 2i hλ(eiϕ1(λ)) , B(λ) = eiσ(λ) 2i hλ(eiϕ2(λ)) , Dν(λ) = e2iσ(λ) sin ϕ(λ) (uν(λ) − eiϕ1(λ))(uν(λ) − eiϕ2(λ))h λ(uν(λ)) , Fν(λ) = sin ϕ(λ) (uν(λ) − e−iϕ1(λ))(uν(λ) − e−iϕ2(λ))h λ(uν(λ)) × |hλ(eiϕ1(λ) )hλ(eiϕ2(λ) )| hλ(eiϕ1(λ))hλ(eiϕ2(λ)) and define the vector w (n) j (λ) = (w (n) j,m(λ))n m=1 by w (n) j,m(λ) = A(λ)e−imϕ1(λ) − B(λ)e−imϕ2(λ) + r−1 ν=1 Dν(λ) 1 uν(λ)m + Fν(λ) (−1)j+1 e−i(n+1)σ(λ) uν(λ)n+1−m . The assumption that zeros uν(λ) are all simple guarantees that h (uν) = 0. We denote by · 2 and · ∞ the 2 and ∞ norms on Cn , respectively. Here are our main results. Theorem 1.1. As n → ∞ and if λ (n) j ∈ (α, β), w (n) j (λ (n) j )2 2 = n 4 1 |hλ(eiϕ1(λ))|2 + 1 |hλ(eiϕ2(λ))|2 λ=λ (n) j + O(1), uniformly in j. Theorem 1.2. Let n → ∞ and suppose λ (n) j ∈ (α, β). Then the eigenvectors v (n) j are of the form v (n) j = τ (n) j w (n) j (λ (n) j ) w (n) j (λ (n) j )2 + O∞(e−δn ) where τ (n) j ∈ T and O∞(e−δn ) denotes vectors ξ (n) j ∈ Cn such that ξ (n) j ∞ ≤ Ke−δn for all j and n with some finite constant K independent of j and n. Note that the previous theorem gives (1.2) with Ωm(λ) = A(λ)e−imϕ1(λ) − B(λ)e−imϕ2(λ) , Φm(λ) = r−1 ν=1 Dν(λ) uν(λ)m , e−iη(λ) = |hλ(eiϕ1(λ) )hλ(eiϕ2(λ) )| hλ(eiϕ1(λ))hλ(eiϕ2(λ)) .
- 31. 20 A. Böttcher, S.M. Grudsky and E.A. Maksimenko Things can be a little simplified for symmetric matrices. Thus, suppose all ak are real and ak = a−k for all k. We will show that then {u1(λ), . . . , ur−1(λ)} = {u1(λ), . . . , ur−1(λ)}. Put Qν(λ) = |hλ(eiϕ(λ) )| sin ϕ(λ) (uν(λ) − eiϕ(λ))(uν(λ) − e−iϕ(λ))h λ(uν(λ)) and let y (n) j (λ) = (y (n) j,m(λ))n m=1 be given by y (n) j,m(λ) = sin mϕ(λ) + θ(λ) 2 − r−1 ν=1 Qν(λ) 1 uν(λ)m + (−1)j+1 uν(λ)n+1−m . (1.6) Theorem 1.3. Let n → ∞ and suppose λ (n) j ∈ (α, β). If ak = a−k for all k, then y (n) j (λ (n) j )2 2 = n 2 + O(1) uniformly in j, and the eigenvectors v (n) j are of the form v (n) j = τ (n) j y (n) j (λ (n) j ) y (n) j (λ (n) j )2 + O∞(e−δn ) where τ (n) j ∈ T and O∞(e−δn ) is as in the previous theorem. Let J be the n × n matrix with ones on the counterdiagonal and zeros else- where. Thus, (Jv)m = vn+1−m. A vector v is called symmetric if Jv = v and skew-symmetric if Jv = −v. Trench [17] showed that the eigenvectors v (n) 1 , v (n) 3 , . . . are all symmetric and that the eigenvectors v (n) 2 , v (n) 4 , . . . are all skew-symmetric. From (1.5) we infer that sin (n + 1 − m)ϕ(λ (n) j ) + θ(λ (n) j ) 2 = (−1)j+1 sin mϕ(λ (n) j ) + θ(λ (n) j ) 2 + O(e−δn ) and hence (1.6) implies that (Jy (n) j (λ (n) j ))m = (−1)j+1 y (n) j,m(λ (n) j ) + O(e−δn ). Consequently, apart from the term O(e−δn ), the vectors y (n) j (λ (n) j ) are symmetric for j = 1, 3, . . . and skew-symmetric for j = 2, 4, . . .. This is in complete accordance with Trench’s result. Due to (1.5), we also have sin mϕ(λ (n) j ) + θ(λ (n) j ) 2 = sin m − n + 1 2 ϕ(λ (n) j ) + O(e−δn ).
- 32. Eigenvectors of Large Hermitian Toeplitz Band Matrices 21 Thus, Theorem 1.3 remains valid with (1.6) replaced by y (n) j,m(λ) = sin m − n + 1 2 ϕ(λ) + πj 2 − r−1 ν=1 Qν(λ) 1 uν(λ)m + (−1)j+1 uν(λ)n+1−m . (1.7) In this expression, the function θ has disappeared. Define y (n) j again by (1.6). The following theorem in conjunction with Theo- rem 1.3 proves (1.3). Theorem 1.4. Let n → ∞ and suppose λ (n) j ∈ (α, β). If ak = a−k for all k, then y (n) j (λ (n) j ) y (n) j (λ (n) j )2 − x (n) j 2 = O j n . The rest of the paper is as follows. We approach eigenvectors by using the elementary observation that if λ is an eigenvalue of Tn(a), then every nonzero column of the adjugate matrix of Tn(a) − λI = Tn(a − λ) is an eigenvector for λ. In Section 2 we employ “exact” formulas by Trench and Widom for the inverse and the determinant of a banded Toeplitz matrix to get a representation of the first column of the adjugate matrix of Tn(a − λ) that will be convenient for asymptotic analysis. This analysis is carried out in Section 3. On the basis of these results, Theorems 1.1 and 1.2 are proved in Section 4, while the proofs of Theorems 1.3 and 1.4 are given in Section 4. Section 6 contains numerical results. 2. The first column of the adjugate matrix The adjugate matrix adj B of an n × n matrix B = (bjk)n j,k=1 is defined by (adj B)jk = (−1)j+k det Mkj where Mkj is the (n − 1) × (n − 1) matrix that results from B by deleting the kth row and the jth column. We have (A − λI) adj (A − λI) = (det(A − λI))I. Thus, if λ is an eigenvalue of A, then each nonzero column of adj (A − λI) is an eigenvector. For an invertible matrix B, adj B = (det B)B−1 . (2.1) Formulas for det Tn(b) and T −1 n (b) were established by Widom [18] and Trench [16], respectively. The purpose of this section is to transform Trench’s formula for the first column of T −1 n (b) into a form that will be convenient for further analysis.
- 33. 22 A. Böttcher, S.M. Grudsky and E.A. Maksimenko Theorem 2.1. Let b(t) = q k=−p bktk = bpt−q p+q j=1 (t − zj) (t ∈ T) where p ≥ 1, q ≥ 1, bp = 0, and z1, . . . , zp+q are pairwise distinct nonzero complex numbers. If n p + q and 1 ≤ m ≤ n, then the mth entry of the first column of adj Tn(b) is [adj Tn(b)]m,1 = J⊂Z,|J|=p CJ Wn J z∈J Sm,J,z (2.2) where Z = {z1, . . . , zp+q}, the sum is over all sets J ⊂ Z of cardinality p, and, with J := Z J, CJ = z∈J zq z∈J,w∈J 1 z − w , WJ = (−1)p bp z∈J z, Sm,J,z = − 1 bp 1 zm w∈J{z} 1 z − w . Proof. It suffices to prove (2.2) under the assumption that det Tn(b) = 0 because both sides of (2.2) are continuous functions of z1, . . . , zp+q. Thus, let det Tn(b) = 0. We will employ (2.1) with B = Tn(b). Trench [16] proved that [T −1 n (b)]m,1 equals − 1 bp D{1,...,p+q}(0, . . . , q − 1, q + n, . . . , q + n + p − 2, q + n − m) D{1,...,p+q}(0, . . . , q − 1, q + n, . . . , q + n + p − 1) (2.3) where D{j1,...,jk}(s1, . . . , sk) denotes the determinant det ⎛ ⎜ ⎜ ⎜ ⎝ zs1 j1 zs2 j1 . . . zsk j1 zs1 j2 zs2 j2 . . . zsk j2 . . . . . . . . . zs1 jk zs2 jk . . . zsk jk ⎞ ⎟ ⎟ ⎟ ⎠ . Note that DJ (s1 + ξ, . . . , sk + ξ) = ⎛ ⎝ j∈J zξ j ⎞ ⎠ DJ (s1, . . . , sk), D{1,2,...,k}(0, 1, . . . , k − 1) = j,∈J j (z − zj).
- 34. Eigenvectors of Large Hermitian Toeplitz Band Matrices 23 We first consider the denominator of (2.3). Put Z = {1, . . ., p + q}. Laplace expansion along the last p columns gives DZ (0, . . . , q − 1, q + n, . . . , q + n + p − 1) = J⊂Z,|J|=p (−1)inv(J,J) DJ (q + n, . . . , q + n + p − 1)DJ (0, . . . , q − 1) = J⊂Z,|J|=p (−1)inv(J,J) k∈J zq+n k k,∈J k (z − zk) k,∈J k (z − zk), where inv(J, J) is the number of inversions in the permutation of length p + q whose first q elements are the elements of the set J in increasing order and whose last p elements are the elements of the set J in increasing order. A little thought reveals that inv(J, J) is just the number of pairs (k, ) with k ∈ J, ∈ J, k . We have j∈J,s∈J (zj − zs) = ∈J,k∈J k (z − zk) k∈J,∈J k (zk − z) = (−1)inv(J,J) ∈J,k∈J k (z − zk) ∈J,k∈J k (z − zk) (2.4) and hence the denominator is equal to Rn J⊂Z,|J|=p CJ Wn J with Rn := (−1)pn bn p k (z − zk). A formula by Widom [18], which can also be found in [1], says that det Tn(b) = J⊂Z,|J|=p CJ Wn J . Consequently, the denominator of (2.3) is nothing but Rn det Tn(b). Let us now turn to the numerator of (2.3). This time Laplace expansion along the last p columns yields DZ(0, . . . , q − 1, q + n, . . . , q + n + p − 2, q + n − m) = J⊂Z,|J|=p (−1)inv(J,J) DJ (q + n, . . . , q + n + p − 1, q + n − m)DJ (0, . . . , q − 1) = J⊂Z,|J|=p (−1)inv(J,J) DJ (0, . . . , q − 1) ⎛ ⎝ j∈J zq+n j ⎞ ⎠ DJ (0, . . . , p − 2, −m). Expanding DJ (0, . . . , p − 2, −m) by its last column we get DJ (0, . . . , p − 2, −m) = j∈J (−1)inv(J{j},j) z−m j DJ{j}(0, . . . , p − 2) (2.5)
- 35. 24 A. Böttcher, S.M. Grudsky and E.A. Maksimenko with inv(J {j}, j) being the number of s ∈ J {j} such that s j. Thus, (2.5) is j∈J (−1)inv(J{j},j) z−m j k,∈J{j} k (z − zk) = j∈J z−m j k,∈J k (z − zk) s∈J{j} 1 zj − zs . This in conjunction with (2.4) shows that the numerator of (2.2) equals −bpRn J⊂Z,|J|=p CJ Wn J z∈J Sm,J,z. In summary, from (2.3) we obtain that [T −1 n (b)]m,1 = 1 det Tn(b) J⊂Z,|J|=p CJ Wn J z∈J Sm,J,z, which after multiplication by det Tn(b) becomes (2.2). 3. The main terms of the first column We now apply Theorem 2.1 to b(t) = a(t) − λ = art−r 2r k=1 (t − zk(λ)) (3.1) where λ ∈ (α, β). The set Z = Z(λ) is given by (1.4). Let d0(λ) = (−1)r areiσ(λ) r−1 k=1 uk(λ). (3.2) In [2], we showed that d0(λ) 0 for all λ ∈ (0, M). The dependence on λ will henceforth frequently be suppressed in notation. Let J1 = {u1, . . . , ur−1, eiϕ1 }, J2 = {u1, . . . , ur−1, eiϕ2 } and for ν ∈ {1, . . . , r − 1}, put J0 ν = {u1, . . . , ur−1, 1/uν}. Lemma 3.1. If J ⊂ Z, |J| = r, J / ∈ {J1, J2, J0 1 , . . . , J0 r−1}, then |CJ Wn j Sm,J,z| ≤ K dn 0 sin ϕ e−δn for all z ∈ J, n ≥ 1, 1 ≤ m ≤ n, λ ∈ (α, β) with some finite constant K that does not depend on z, n, m, λ.
- 36. Eigenvectors of Large Hermitian Toeplitz Band Matrices 25 Proof. If both eiϕ1 and eiϕ2 belong to J, then J = {uν1 , . . . , uνk , eiϕ1 , eiϕ2 , 1/us1 , . . . , 1/us } with k + = r − 2. Since min λ∈[α,β] min j1=j2 |uj1 (λ) − uj2 (λ)| 0, we conclude that |CJ | ≤ K1. Here and in the following Ki denotes a finite constant that is independent of λ ∈ [α, β]. We have k ≤ r − 2 and thus |WJ | = |ar| |uν1 . . . uνk | |us1 . . . us | ≤ d0e−δ |us1 . . . us | . (3.3) If z ∈ {uν1 , . . . , uνk , eiϕ1 , eiϕ2 }, then obviously |Sm,J,z| ≤ K2/ sin ϕ and hence |CJ Wn J Sm,J,z| ≤ K1K2 dn 0 e−δn sin ϕ . In case z ∈ {1/us1 , . . . , 1/us }, say z = 1/us1 , we have |Sm,J,z| ≤ K3|uν1 |m , which gives |CJ Wn J Sm,J,z| ≤ K1K3dn 0 e−δn |uν1 |m |uν1 |n ≤ K1K3dn 0 e−δn ≤ K1K3 dn 0 e−δn sin ϕ . The only other possibility for J is to be of the type J = {uν1 , . . . , uνk , eiϕ1 , 1/us1 , . . . , 1/us } with k + ≤ r − 1, k ≤ r − 2, ≥ 1. (The case where eiϕ1 is replaced by eiϕ2 is completely analogous.) This time, |CJ | ≤ K4/ sin ϕ and (3.3) holds again. For z ∈ {uν1 , . . . , uνk , eiϕ1 } we have |Sm,J,z| ≤ K5 and thus get the assertion. If z = 1/us for some s ∈ {s1, . . . , s}, say s = s1, then |Sm,J,z| ≤ K6|us1 |m , and the assertion follows as above, too. Let d1(λ) = 1 |hλ(eiϕ1(λ))hλ(eiϕ2(λ))| r−1 k,s=1 1 − 1 uk(λ)us(λ) −1 . It is easily seen that d1(λ) 0 for all λ ∈ [0, M]. Lemma 3.2. If λ = λ (n) j ∈ (0, M), then CJ1 Wn J1 Sm,J1,eiϕ1 = d1dn−1 0 sin ϕ (−1)j Ae−imϕ1 + O(e−δn ) , CJ2 Wn J2 Sm,J2,eiϕ2 = d1dn−1 0 sin ϕ (−1)j+1 Be−imϕ2 + O(e−δn ) uniformly in m and λ.
- 37. 26 A. Böttcher, S.M. Grudsky and E.A. Maksimenko Proof. We abbreviate r−1 k=1 to k. Clearly, Wj1 = (−1)r ar k uk eiϕ1 = (−1)r ar k uk eiσ eiϕ = d0eiϕ . We have CJ1 = ( k ur k) eirϕ1 (eiϕ1 − eiϕ2 ) k,s uk − 1 us k (uk − eiϕ2 ) k eiϕ1 − 1 uk = eirϕ1 eiσ(eiϕ − e−iϕ) k,s 1 − 1 ukus k 1 − eiϕ2 uk ei(r−1)ϕ1 k 1 − e−iϕ1 uk = eiϕ 2i sin ϕ k,s 1 − 1 ukus h(eiϕ2 )h(eiϕ1 ) and because h(eiϕ2 )h(eiϕ1 ) = |h(eiϕ1 )h(eiϕ2 )|e−iθ , (3.4) it follows that CJ1 = d1ei(ϕ+θ) 2i sin ϕ . Furthermore, Sm,J1,eiϕ1 = − 1 ar 1 eimϕ1 k(eiϕ1 − uk) = − 1 ar e−imϕ1 (−1)r−1 ( k uk) k(1 − eiϕ1 /uk) = e−im(σ+ϕ) (−1)rar ( k uk) h(eiϕ1 ) = e−im(σ+ϕ) eiσ d0h(eiϕ1 ) . Putting things together we arrive at the formula CJ1 Wn J1 Sm,J1,eiϕ1 = d1dn−1 0 sin ϕ A e−im(σ+ϕ) ei((n+1)ϕ+θ) . Obviously, σ + ϕ = ϕ1. By virtue of (1.5), ei((n+1)ϕ+θ) = eiπj (1 + O(e−δn )) = (−1)j (1 + O(e−δn )). This proves the first of the asserted formulas. Analogously, WJ2 = d0e−iϕ , CJ2 = − d1e−i(ϕ+θ) 2i sin ϕ , Sm,J2,eiϕ2 = e−im(σ−ϕ) eiσ d0h(eiϕ2 ) , which gives the second formula. Lemma 3.3. If 1 ≤ ν ≤ r − 1 and λ = λ (n) j ∈ (α, β), then, uniformly in m and λ, CJ1 Wn J1 Sm,J1,uν + CJ2 Wn J2 Sm,J2,uν = d1dn−1 0 sin ϕ (−1)j Dν 1 um ν + O(e−δn ) .
- 38. Eigenvectors of Large Hermitian Toeplitz Band Matrices 27 Proof. By definition, Sm,J1,uν = − 1 ar 1 um ν (uν − eiϕ1 ) s=ν(uν − us) = u−m ν (−1)r−1 ( k uk) ar(uν − eiϕ1 )h(uν)) Since −h (z) equals 1 u1 1 − z u2 . . . 1 − z ur−1 + · · · + 1 ur−1 1 − z u1 . . . 1 − z ur−2 , we obtain that h (uν) = − 1 uν s=ν 1 − uν us . Thus, Sm,J1,uν = u−m ν (−1)rar ( k uk) (uν − eiϕ1 )h(uν) = u−m ν eiσ d0(uν − eiϕ1 )h(uν) . Changing ϕ1 to ϕ2 we get Sm,J2,uν = u−m ν eiσ d0(uν − eiϕ2 )h(uν) . These two expressions along with the expressions for CJ1 , WJ1 , CJ2 , WJ2 derived in the proof of Lemma 3.2 show that the sum under consideration is d1dn−1 0 2i sin ϕ u−m ν eiσ h(uν) ei((n+1)ϕ+θ) uν − eiϕ1 − e−i((n+1)ϕ+θ) uν − eiϕ2 . Because of (1.5), the term in brackets equals (−1)j 1 uν − eiϕ1 − 1 uν − eiϕ2 + O(e−δn ) = (−1)j eiσ 2i sinϕ (uν − eiϕ1 )(uν − eiϕ2 ) + O(e−δn ). Lemma 3.4. For 1 ≤ ν ≤ r − 1 and λ ∈ (α, β), CJ0 ν Wn J0 ν Sm,J0 ν ,1/uν = − d1dn−1 0 sin ϕ Fν e−i(n+1)σ un+1−m ν .
- 39. 28 A. Böttcher, S.M. Grudsky and E.A. Maksimenko Proof. We have CJ0 ν = ( k ur k) /(ur νP1P2P3) with P1 = 1 uν − eiϕ1 1 uν − eiϕ2 = (uν − e−iϕ1 )(uν − e−iϕ2 ) u2 νe−2iσ , P2 = k (uk − eiϕ1 ) k (uk − eiϕ2 ) = k u2 k h(eiϕ1 )h(eiϕ2 ), P3 = s=ν 1 uν − 1 us k s=ν uk − 1 us = 1 ur−2 ν s=ν 1 − uν us k ur−2 k k s=ν 1 − 1 ukus = − 1 ur−3 ν h(uν) k ur−2 k 1 d1|h(eiϕ1 )h(eiϕ2 )| 1 k(1 − 1/(ukuν)) . Thus, CJ0 ν equals − d1e−2iσ |h(eiϕ1 )h(eiϕ2 )| uν(uν − e−iϕ1 )(uν − e−iϕ2 )h(uν)h(eiϕ1 )h(eiϕ2 ) k 1 − 1 ukuν . Since WJ0 ν = d0e−iσ /uν and Sm,J0 ν ,1/uν = − 1 ar um ν 1 k(1/uν − uk) = um ν (−1)rar ( k uk) k 1 − 1 ukuν = um ν eiσ d0 k 1 − 1 ukuν , we obtain that CJ0 ν Wn J0 ν Sm,J0 ν ,1/uν is equal to − d1dn−1 0 un+1−m e−iσ e−inσ |h(eiϕ1 )h(eiϕ2 )| (uν − e−iϕ1 )(uν − e−iϕ2 )h(uν)h(eiϕ1 )h(eiϕ2 ) . Lemma 3.5. If 1 ≤ k ≤ r − 1 and λ ∈ (α, β), CJ0 ν Wn J0 ν Sm,J0 ν ,uk = d1dn−1 0 sin ϕ O(e−δn ) uniformly in m and λ.
- 40. Eigenvectors of Large Hermitian Toeplitz Band Matrices 29 Proof. This time Sm,J0 ν ,uk = − 1 ar 1 um k 1 (uk − 1/uν) s=k(uk − us) = u−m k (−1)r−1aruk 1 − 1 ukuν s=k us s=k 1 − uk us = − 1 d0um k 1 1 − 1 ukuν s=k 1 − uk us . Expressions for CJ0 ν and WJ0 ν were given in the proof of Lemma 3.4. It follows that CJ0 ν Wn J0 ν Sm,J0 ν ,uk = Gν,k d1dn−1 0 sin ϕ 1 un+1 ν um k where Gν,k equals e2iσ e−inσ |h(eiϕ1 )h(eiϕ2 )| sin ϕ (uν − e−iϕ1 )(uν − e−iϕ2 )h(uν)h(eiϕ1 )h(eiϕ2 ) s=k 1 − 1 usuν s=k 1 − uk us . Since h(uν) = − 1 uν s=ν 1 − uν us , we see that Gν,k remains bounded on [α, β]. Finally, 1 |un+1 ν um k | ≤ 1 |uν|n ≤ e−δn . Corollary 3.6. If λ = λ (n) j ∈ (α, β), then [adj Tn(a − λ)]m,1 = (−1)j d1(λ)dn−1 0 (λ) sin ϕ(λ) [wj,m(λ) + O(e−δn )] uniformly in m and λ. Proof. This follows from Theorem 2.1 and Lemmas 3.1 to 3.5 along with the fact that d1 is bounded and bounded away from zero on [α, β]. 4. The asymptotics of the eigenvectors We now prove Theorem 1.1. There is a finite constant K1 such that |Dν| ≤ K1 and |Fν| ≤ K1 for all ν and all λ ∈ (α, β). Thus, summing up two finite geometric series, we get n m=1 Dν 1 um ν + Fν (−1)j+1 e−i(n+1)σ un+1−m ν 2 ≤ 2K2 1 1 |uν|2 1 − 1/|uν|2(n+1) 1 − 1/|uν|2 ≤ K2
- 41. 30 A. Böttcher, S.M. Grudsky and E.A. Maksimenko for all ν, n, λ. We further have n m=1 Ae−imϕ1 − Be−imϕ2 2 = n m=1 e−imϕ1 2h(eiϕ1 ) − e−imϕ2 2h(eiϕ2 ) 2 = n m=1 1 4|h(eiϕ1 )|2 + 1 4|h(eiϕ1 )|2 − n m=1 e−2imϕ 4h(eiϕ1 )h(eiϕ2 ) + e2imϕ 4h(eiϕ1 )h(eiϕ2 ) . The first sum is of the form n m=1(γ/4) and therefore equals (n/4)γ. Hence, because of (3.4) we are left to prove that n m=1 eiθ(λ (n) j ) e2imϕ(λ (n) j ) ≤ K3 (4.1) for all n and j such that λ (n) j ∈ (α, β). The sum in (4.1) is ei[(n+1)ϕ(λ (n) j )+θ(λ (n) j )] sin nϕ(λ (n) j ) sin ϕ(λ (n) j ) . Thus, (4.1) will follow as soon as we have shown that sin nϕ(λ (n) j ) sin ϕ(λ (n) j ) ≤ K3 for all n and j in question. From (1.5) we infer that nϕ(λ (n) j ) = πj − ϕ(λ (n) j ) − θ(λ (n) j ) + O(e−δn ), which implies that sin nϕ(λ (n) j ) = (−1)j+1 sin ϕ(λ (n) j ) + θ(λ (n) j ) + O(e−δn ). Suppose first that 0 ϕ(λ (n) j ) ≤ π/2. Then sin ϕ(λ (n) j ) + θ(λ (n) j ) sin ϕ(λ (n) j ) ≤ π 2 |ϕ(λ (n) j ) + θ(λ (n) j )| |ϕ(λ (n) j )| ≤ π 2 1 + |θ(λ (n) j )| |ϕ(λ (n) j )| . (4.2) In [2], we proved that |θ/ϕ| is bounded on (0, M). Thus, the right-hand side of (4.2) is bounded by some K3 for all n and j. If π/2 ϕ(λ (n) j ) π, we may replace (4.2) by the upper bound π 2 1 + |θ(λ (n) j )| |π − ϕ(λ (n) j )| . We know again from [2] that |θ/(π −ϕ)| is bounded on (0, M). This completes the proof of Theorem 1.1.
- 42. Eigenvectors of Large Hermitian Toeplitz Band Matrices 31 Here is the proof of Theorem 1.2. By virtue of Theorem 1.1, wj(λ (n) j )2 1 whenever n is sufficiently large. Corollary 3.6 therefore implies that the first column of adjTn(a − λ (n) j ) is nonzero and thus an eigenvector for λ (n) j for all n ≥ n0 and all 1 ≤ j ≤ n such that λ (n) j ∈ (α, β). Again by Corollary 3.6, the mth entry of this column is d1(λ)dn−1 0 (λ) sin ϕ(λ) [wj,m(λ) + ξ (n) j,m] λ=λ (n) j where |ξ (n) j,m| ≤ Ke−δn for all n and j under consideration and K does not depend on m, n, j. It follows that wj(λ (n) j ) + ξ (n) j,m n m=1 = wj(λ (n) j ) + O∞(e−δn ) is also an eigenvector for λ (n) j . Consequently, wj(λ (n) j ) + O∞(e−δn ) wj(λ (n) j ) + O∞(e−δn)2 = wj(λ (n) j ) wj(λ (n) j )2 + O∞(e−δn ) (4.3) is a normalized eigenvector for λ (n) j . From (1.5) we deduce that all eigenvalues of Tn(a) are simple. Thus, v (n) j is a scalar multiple of modulus 1 of (4.3). This completes the proof of Theorem 1.2. 5. Symmetric matrices The matrices Tn(a) are all symmetric if and only if all ak are real and ak = a−k for all k. Obviously, this is equivalent to the requirement that the real-valued function g(x) := a(eix ) be even, that is, g(x) = g(−x) for all x. Thus, suppose g is even. In that case ϕ0 = π, ϕ1(λ) = −ϕ2(λ) = ϕ(λ), σ(λ) = 0. Moreover, for t ∈ T we have art−r 2r k=1 (t − zk(λ)) = a(t) − λ = a(1/t) − λ = artr 2r k=1 (1/t − zk(λ)) = ar 2r k=1 zk(λ) t−r 2r k=1 (t − 1/zk(λ)), which in conjunction with (1.4) implies that {u1(λ), . . . , ur−1(λ)} = {u1(λ), . . . , ur−1(λ)}. (5.1) The coefficients of the polynomial hλ(t) are symmetric functions of 1/u1(λ), . . . , 1/ur−1(λ).
- 43. 32 A. Böttcher, S.M. Grudsky and E.A. Maksimenko From (5.1) we therefore see that these coefficients are real. It follows in particular that hλ(e−iϕ(λ) ) = hλ(eiϕ(λ)), which gives θ(λ) = 2 arg hλ(eiϕ(λ) ) and thus hλ(eiϕ(λ) ) = |hλ(eiϕ(λ) )|eiθ(λ)/2 , hλ(e−iϕ(λ) ) = |hλ(eiϕ(λ) )|e−iθ(λ)/2 . We are now in a position to prove Theorem 1.3. To do so, we use Theorem 1.2. Consider the vector wj(λ (n) j ). We now have Ae−imϕ1 − Be−imϕ2 = e−imϕ 2ih(eiϕ) − eimϕ 2ih(e−iϕ) = 1 2i|h(eiϕ)| e−imϕ eiθ/2 − eimϕ e−iθ/2 = − 1 |h(eiϕ)| sin mϕ + θ 2 . Furthermore, Dν = sin ϕ (uν − eiϕ)(uν − e−iϕ)h(uν) = Qν |h(eiϕ)| , Fν = sin ϕ (uν − eiϕ)(uν − e−iϕ)h(uν) |h(eiϕ )h(e−iϕ )| h(eiϕ)h(e−iϕ) = sin ϕ (uν − eiϕ)(uν − e−iϕ)h(uν) . Consequently, from (5.1) we infer that r−1 ν=1 Fν un+1−m ν = r−1 ν=1 Dν un+1−m ν = r−1 ν=1 Qν |h(eiϕ)| un+1−m ν . In summary, it follows that wj(λ (n) j ) = − 1 |h(eiϕ(λ (n) j ) )| yj(λ (n) j ). (5.2) Thus, the representation v (n) j = τ (n) j yj(λ (n) j ) yj(λ (n) j )2 + O∞(e−δn ) is immediate from Theorem 1.2. Finally, put hj,n = |h(eiϕ(λ (n) j ) )| = |h(e−iϕ(λ (n) j ) )|. Theorem 1.1 shows that wj(λ (n) j )2 2 = n 4 1 |h(eiϕ)|2 + 1 |h(e−iϕ)|2 λ=λ (n) j + O(1) = n 2h2 j,n + O(1), whence, by (5.2), yj(λ (n) j )2 2 = h2 j,nwj(λ (n) j )2 2 = n/2 + O(1). The proof of The- orem 1.3 is complete. Here is the proof of Theorem 1.4. We first estimate the “small terms” in y (n) j . Summing up finite geometric series and using the assumption that |uν(λ)|
- 44. Eigenvectors of Large Hermitian Toeplitz Band Matrices 33 are separated from 1 we come to n m=1 r−1 ν=1 Qν(λ) 1 uν(λ)m + (−1)j+1 uν(λ)n+1−m 2 ≤ r−1 ν=1 4(r − 1)|Qν(λ)|2 1 − |uν(λ)|2 ≤ K sin2 ϕ(λ) where K is some positive number depending only on a. since ϕ(λ (n) j ) = O (j/n), it follows that r−1 ν=1 Qν(λ) 1 uν(λ)m + (−1)j+1 uν(λ)n+1−m n m=1 2 = O j n . (5.3) We next consider the difference between the “main term” of y (n) j and sin mjπ n+1 . Using the elementary estimate | sin A − sin B|2 = 4 sin2 A − B 2 cos2 A + B 2 ≤ 4 sin2 A − B 2 = 2 − 2 cos(A − B), we get n m=1 sin mϕ(λ (n) j ) + θ(λ (n) j ) − sin mjπ n + 1 2 ≤ 2n − 2 n m=1 cos m ϕ(λ (n) j ) − πj n + 1 + θ(λ (n) j ) . To simplify the last sum, we use that n m=1 cos(mξ + ω) = sin nξ 2 cos (n+1)ξ 2 + ω sin ξ 2 = n 1 + O(n2 ξ2 ) 1 + O (n + 1)ξ 2 + ω 2 . In our case ω = θ(λ (n) j ) = O λ (n) j = O j n , ξ = ϕ(λ (n) j ) − πj n + 1 = − θ(λ (n) j ) n + 1 + O(e−nδ ) = O j n2 . Consequently, n m=1 sin mϕ(λ (n) j ) + θ(λ (n) j ) − sin mjπ n + 1 2 = O j2 n ,
- 45. 34 A. Böttcher, S.M. Grudsky and E.A. Maksimenko that is, sin mϕ(λ (n) j ) + θ(λ (n) j ) − sin mjπ n + 1 n m=1 2 = O j √ n . (5.4) Combining (5.3) and (5.4)we obtain that y (n) j − n + 1 2 x (n) j 2 = O j n + O j √ n = O j √ n , (5.5) which implies in particular that y (n) j 2 = n + 1 2 1 + O j n . (5.6) Clearly, estimates (5.5) and (5.6) yield the asserted estimate. This completes the proof of Theorem 1.4. 6. Numerical results Given Tn(a), determine the approximate eigenvalue λ (n) j,∗ from the equation (n + 1)ϕ(λ (n) j,∗ ) + θ(λ (n) j,∗ ) = πj. In [2], we proposed an exponentially fast iteration method for solving this equation. Let w (n) j (λ) ∈ Cn be as in Section 1 and put w (n) j,∗ = w (n) j (λ (n) j,∗ ) w (n) j (λ (n) j,∗ )2 . We define the distance between the normalized eigenvector v (n) j and the normalized vector w (n) j,∗ by (v (n) j , w (n) j,∗ ) := min τ∈T τv (n) j − w (n) j,∗ 2 = 2 − 2 v (n) j , w (n) j,∗ and put Δ (n) ∗ = max 1≤j≤n |λ (n) j − λ (n) j,∗ |, Δ(n) v,w = max 1≤j≤n (v (n) j , w (n) j,∗ ), Δ(n) r = max 1≤j≤n Tn(a)w (n) j,∗ ) − λ (n) j,∗ w (n) j,∗ 2. The tables following below show these errors for three concrete choices of the generating function a.
- 46. Eigenvectors of Large Hermitian Toeplitz Band Matrices 35 For a(t) = 8 − 5t − 5t−1 + t2 + t−2 we have n = 10 n = 20 n = 50 n = 100 n = 150 Δ (n) ∗ 5.4 · 10−7 1.1 · 10−11 5.2 · 10−25 1.7 · 10−46 9.6 · 10−68 Δ (n) v,w 2.0 · 10−6 1.1 · 10−10 2.0 · 10−23 1.9 · 10−44 2.0 · 10−65 Δ (n) r 8.0 · 10−6 2.7 · 10−10 3.4 · 10−23 2.2 · 10−44 1.9 · 10−65 If a(t) = 8 + (−4 − 2i)t + (−4 − 2i)t−1 + it − it−1 then n = 10 n = 20 n = 50 n = 100 n = 150 Δ (n) ∗ 3.8 · 10−8 2.8 · 10−13 2.9 · 10−30 5.9 · 10−58 1.6 · 10−85 Δ (n) v,w 1.8 · 10−7 4.7 · 10−13 2.0 · 10−29 7.0 · 10−57 2.4 · 10−84 Δ (n) r 5.4 · 10−7 1.3 · 10−12 2.7 · 10−29 6.7 · 10−57 1.9 · 10−84 In the case where a(t) = 24 + (−12 − 3i)t + (−12 + 3i)t−1 + it3 − it−3 we get n = 10 n = 20 n = 50 n = 100 n = 150 Δ (n) ∗ 6.6 · 10−6 1.2 · 10−10 7.6 · 10−24 1.4 · 10−45 3.3 · 10−67 Δ (n) v,w 1.9 · 10−6 1.3 · 10−10 2.0 · 10−23 7.2 · 10−45 2.8 · 10−66 Δ (n) r 2.5 · 10−5 8.6 · 10−10 7.3 · 10−23 1.9 · 10−44 5.9 · 10−66 References [1] A. Böttcher and S.M. Grudsky, Spectral Properties of Banded Toeplitz Matrices, SIAM, Philadelphia 2005. [2] A. Böttcher, S.M. Grudsky, E.A. Maksimenko, Inside the eigenvalues of certain Her- mitian Toeplitz band matrices. Journal of Computational and Applied Mathematics 233 (2010), 2245–2264. [3] A. Böttcher, S.M. Grudsky, E.A. Maksimenko, and J. Unterberger, The first or- der asymptotics of the extreme eigenvectors of certain Hermitian Toeplitz matrices. Integral Equations and Operator Theory 63 (2009), 165–180. [4] A. Böttcher, S.M. Grudsky, and J. Unterberger, Asymptotic pseudomodes of Toeplitz matrices, Operators and Matrices 2 (2008), 525–541. [5] A. Böttcher and B. Silbermann, Introduction to Large Truncated Toeplitz Matrices, Universitext, Springer-Verlag, New York 1999. [6] C. Estatico and S. Serra Capizzano, Superoptimal approximation for unbounded sym- bols, Linear Algebra Appl. 428 (2008), 564–585. [7] U. Grenander and G. Szegö, Toeplitz Forms and Their Applications, University of California Press, Berkeley 1958. [8] C.M. Hurvich and Yi Lu, On the complexity of the preconditioned conjugate gradient algorithm for solving Toeplitz systems with a Fisher-Hartwig singularity, SIAM J. Matrix Anal. Appl. 27 (2005), 638–653. [9] M. Kac, W.L. Murdock, and G. Szegö, On the eigenvalues of certain Hermitian forms, J. Rational Mech. Anal. 2 (1953), 767–800.
- 47. 36 A. Böttcher, S.M. Grudsky and E.A. Maksimenko [10] A.Yu. Novosel’tsev and I.B. Simonenko, Dependence of the asymptotics of extreme eigenvalues of truncated Toeplitz matrices on the rate of attaining an extremum by a symbol, St. Petersburg Math. J. 16 (2005), 713–718. [11] S.V. Parter, Extreme eigenvalues of Toeplitz forms and applications to elliptic dif- ference equations, Trans. Amer. Math. Soc. 99 (1961), 153–192. [12] S.V. Parter, On the extreme eigenvalues of Toeplitz matrices, Trans. Amer. Math. Soc. 100 (1961), 263–276. [13] S. Serra Capizzano, On the extreme spectral properties of Toeplitz matrices generated by L1 functions with several minima/maxima, BIT 36 (1996), 135–142. [14] S. Serra Capizzano, On the extreme eigenvalues of Hermitian (block ) Toeplitz ma- trices, Linear Algebra Appl. 270 (1998), 109–129. [15] S. Serra Capizzano and P. Tilli, Extreme singular values and eigenvalues of non- Hermitian block Toeplitz matrices, J. Comput. Appl. Math. 108 (1999), 113–130. [16] W.F. Trench, Explicit inversion formulas for Toeplitz band matrices, SIAM J. Al- gebraic Discrete Methods 6 (1985), 546–554. [17] W.F. Trench, Interlacement of the even and odd spectra of real symmetric Toeplitz matrices, Linear Algebra Appl. 195 (1993), 59–68. [18] H. Widom, On the eigenvalues of certain Hermitian operators, Trans. Amer. Math. Soc. 88 (1958), 491–522. [19] N.L. Zamarashkin and E.E. Tyrtyshnikov, Distribution of the eigenvalues and sin- gular numbers of Toeplitz matrices under weakened requirements on the generating function, Sb. Math. 188 (1997), 1191–1201. [20] P. Zizler, R.A. Zuidwijk, K.F. Taylor, and S. Arimoto, A finer aspect of eigenvalue distribution of selfadjoint band Toeplitz matrices, SIAM J. Matrix Anal. Appl. 24 (2002), 59–67. Albrecht Böttcher Fakultät für Mathematik TU Chemnitz D-09107 Chemnitz, Germany e-mail: aboettch@mathematik.tu-chemnitz.de Sergei M. Grudsky Departamento de Matemáticas, CINVESTAV del I.P.N. Apartado Postal 14-740 07000 México, D.F., México e-mail: grudsky@math.cinvestav.mx Egor A. Maksimenko Departamento de Matemáticas, CINVESTAV del I.P.N. Apartado Postal 14-740 07000 México, D.F., México e-mail: emaximen@math.cinvestav.mx
- 48. Operator Theory: Advances and Applications, Vol. 210, 37–42 c 2010 Springer Basel AG Complete Quasi-wandering Sets and Kernels of Functional Operators Victor D. Didenko To Nikolai Vasilevski on the occasion of his 60th birthday Abstract. Kernels of functional operators generated by mapping that possess complete quasi-wandering sets are studied. It is shown that the kernels of the operators under consideration either consist of a zero element or contain a subset isomorphic to a space L∞(S), where S ⊂ Rn has a positive Lebesgue measure. Consequently, such operators are Fredholm if and only if they are invertible. Mathematics Subject Classification (2000). Primary 47B33, 39A33, 45E10; Sec- ondary 42C40, 39B42. Keywords. Quasi-wandering set, homogeneous equation, solution. 1. Introduction Let X be a domain in Rn provided with the Lebesgue measure μ, and let α : X → X be a measurable mapping satisfying the compatibility condition. Thus if E is a Lebesgue measurable subset of X and if μ(E) = 0, then μ(α−1 (E)) = 0. Assume that A0, A1, . . . , As : X → Cd×d are matrix functions, and consider the functional equation A0(x)ϕ(x) + A1(x)ϕ(α(x)) + · · · + As(x)ϕ(αs (x)) = f(x) (1.1) where ϕ : X → Cd is an unknown vector function and f : X → Cd is a given vector function. Equations of the form (1.1) arise in various fields of mathematics and its applications, and there is vast literature where different properties of such equations and the corresponding operators are studied. For details, the reader can consult [1, 2, 4, 5, 9, 10, 12] and literature therein. Let us mention a few functional operators relevant to our following discussion. Let A = A(x), x ∈ Rn be a matrix function and M ∈ Rn×n a non-singular integer expansive matrix – i.e., all eigenvalues λj, j = 1, 2, . . ., n of the matrix M satisfy
- 49. 38 V.D. Didenko the inequality |λj| 1. Equations of the form ϕ(x) = A(x)ϕ(Mx) (1.2) appear in most publications concerning wavelets or their applications. In particu- lar, equation (1.2) arises in discussions of the discrete refinement equation ψ(x) = k∈Zn akψ(Mx − k), x ∈ Rn , (1.3) or the continuous refinement equation ψ(x) = Rn c(x − My)ψ(x) dy, x ∈ Rn . (1.4) Note that the refinement equation (1.3) is associated with the functional equa- tion (1.2), where the matrix A has a uniformly convergent Fourier series; whereas equation (1.4) leads to the equation (1.2), with the matrix A that is the Fourier image of the matrix c ∈ Ld×d 1 (Rn ). Let us also recall another class of functional equations. Consider X = Rn and choose an h ∈ Rn , h = 0. The difference equations A0(x)ϕ(x) + A1(x)ϕ(x + h) + · · · + As(x)ϕ(x + sh) = f(x), that often arise in applications have been investigated by various authors. In the present paper, the kernel spaces and the Fredholm properties of the equations mentioned are studied from a unified point of view. Let Uα := s j=0 AjT j α (1.5) be the operator defined by the left-hand side of equation (1.1), where Tαϕ(x) := ϕ(α(x)), and Aj, j = 0, 1, . . . , s are the operators of multiplication by the matrices Aj(x). In the present paper, the operator Uα is considered on the space Ld 2(X) of all Lebesgue measurable square summable vector functions. We study the kernel space of the functional operators for mappings α that possess the so-called complete quasi-wandering set. For such mappings, the kernel space of the above-mentioned operator Uα has a distinctive property – viz. it either consists of the single element ϕ0 = 0 or it contains a subset isomorphic to a space L∞(S), where S is a subset of Rn with a positive Lebesgue measure. Note that it was conjectured in [3] that the kernel of any operator (1.5) on Lp space is either infinite-dimensional or only has the zero element. However, an example from [11] shows that this conjecture is not true in general. On the other hand, the problem mentioned is closely connected to the hypothesis that any operator (1.5) is Fredholm if and only if it is invertible. Although valid for a number of operator algebras [1, 8], this hypothesis is also not true in general [6]. Thus, this paper presents another class of operators when both hypotheses are true. Moreover, our approach shows that the kernel space of the operators under consideration can include subspaces as ‘massive’ as L∞(S).
- 50. Complete Quasi-wandering Sets 39 2. The kernel space of the operator Uα Now let us assume that the mapping α : X → X is invertible, and the inverse mapping α−1 also satisfies the compatibility condition. Definition 2.1. A Lebesgue measurable set E ⊂ X is a quasi-wandering set for the mapping α if, for any j, k ∈ Z, j = k, either αj (E) ∩ αk (E) = ∅, or if αj1 (E) ∩ αk1 (E) = ∅ for some j1, k1 ∈ Z, then αj1 (x) = αk1 (x) for all x ∈ E. If αj (E) ∩ αk (E) = ∅ for all j, k ∈ Z, j = k, the set E is called the wandering set for α. Definition 2.2. A quasi-wandering set E is called complete if X = ! j∈Z αj (E). It is clear that if a mapping α : X → X possesses a complete quasi-wandering set E and for some indices j, k, j = k, αj (x) = αk (x) for all x ∈ E, then X = N ! j=0 αj (E) for an N ∈ N. Let us now consider a few mappings with complete quasi-wandering sets. Example 1. Suppose X = R and α : R → R is the shift operator α(x) := x − 1. The mapping α satisfies all of the conditions mentioned. It is invertible and the interval [0, 1) is a complete wandering set for α. Example 2. Let X = R2 , and let Λ be a diagonal matrix Λ = λ1 0 0 λ2 , where λ1 = 0 and λ2 1. Consider the mapping αΛ(x) := Λx. Then the set EΛ := (x, y) ∈ R2 , x ∈ R and y ∈ [−1, −1/λ2) ∪ (1/λ2, 1] # is a complete wandering set for the mapping αΛ. Example 3. Let X = R3 , m ∈ N, m ≥ and let αR(x) := Rx, where R is the rotation matrix R = ⎛ ⎝ 1 0 0 0 cos(π/m) − sin(π/m) 0 sin(π/m) cos(π/m) ⎞ ⎠ .
- 51. 40 V.D. Didenko Then the set ER := (x, y, z) ∈ R3 : x ∈ R, y 0, z ≥ 0, and 0 ≤ arctan(z/y) π/m # is a complete quasi-wandering set for the mapping αR and R3 = 2m−1 ! j=0 αj R(ER). Example 4. Let X = E1 ∪E2, E1 ∩E2 = ∅ and μ(E1) = 0 and μ(E2) = 0. Consider a homomorphism α : X → X satisfying the compatibility condition and such that • α(E1) = E2, α(E2) = E1. • α2 (x) = x for all x ∈ X. Then E1 and E2 are complete quasi-wandering sets for the mapping α. Now we can study the kernel spaces of the operators (1.1) when the corre- sponding mapping α possesses a complete quasi-wandering set. Theorem 2.3. Let A1, A2, . . . , As ∈ Ld×d ∞ (X) and α : X → X be a mapping such that Tα : L2(X) → L2(X) is a continuous operator. If the mapping α has a complete quasi-wandering set, then either ker Uα = 0 or there is a subspace S ⊂ ker Uα and a set Sα ⊂ X, the Lebesgue measure of which is positive such that S is isomorphic to the space L∞(Sα). Proof. Let Eα be a complete quasi-wandering set for the mapping α. Assume that the kernel of the operator Uα contains a non-zero element ϕ0. For simplicity, also suppose that d = 1. Thus X |ϕ0(x)|2 dx 0 , so there is at least one index j0 ∈ Z and an ε 0 such that the set Sα = Sε α := x ∈ αj0 (Eα) : |ϕ0(x)| ≥ ε # has a positive Lebesgue measure. Consider now the space L∞(αj0 (Eα)). For any m ∈ L∞(αj0 (Eα)), such that the restriction of m on the set Sε α is a non-zero element of L∞(Sε α), define an extension $ m of the element m on the whole X by $ m(x) := m(α−j (x)) if x ∈ αj+j0 (Eα). Note that the element $ m is well defined and belongs to the space L∞(X). Moreover, it satisfies the equation $ m(x) = $ m(α(x)), x ∈ X. (2.1) Consider now the element $ mϕ0. Taking into account equation (2.1), one can easily check that $ mϕ0 ∈ ker Uα. It remains to show that $ mϕ0 = 0. However, if the
- 52. Complete Quasi-wandering Sets 41 restriction of $ m on the set Sα is a non-zero element of L∞(Sα), then X | $ m(x)ϕ0(x)|2 dx = ∪j∈Zαj (Eα) | $ m(x)ϕ0(x)|2 dx ≥ αj0 (Eα) | $ m(x)ϕ0(x)|2 dx ≥ Sα | $ m(x)ϕ0(x)|2 dx ≥ ε2 Sα | $ m(x)|2 dx 0. Thus the kernel of the operator Uα contains the subset S := { $ mϕ0 : m ∈ L∞(Sα)} and the proof is complete. The connection between some sets related to the operators of multiplication by expansive matrices M and the kernel spaces of the corresponding refinement operators was first noted in [7]. It turns out that these relations have a universal nature and can be extended to general mappings α having quasi-wandering sets. This allows us to characterize the Fredholm properties of the functional operators. Corollary 2.4. Let Uα : L2(X) → L2(X) be a Φ+ -operator. If the mapping α : X → X possesses a complete quasi-wandering set, then ker Uα = 0. Now consider functional operators generated by diffeomorphisms. Theorem 2.5. Let A1, A2, . . . , As ∈ Ld×d ∞ (X) and let α : X → X be a differentiable mapping such that the Jacobian Jα of α satisfies the inequality 0 r1 ≤ Jα(x) ≤ r2, r1, r2 ∈ R (2.2) for all x ∈ X. If the mapping α has a complete quasi-wandering set, then the operator Uα : Ld 2(X) → Ld 2(X) is Fredholm if and only if it is invertible. Proof. If α has a complete quasi-wandering set E, then E is also a complete quasi- wandering set for the inverse mapping α−1 . The proof of Theorem 2.5 therefore follows from Theorem 2.3 and from condition (2.2), which implies that the struc- ture of the adjoint operator U∗ α is similar to the structure of the operator Uα. Remark 2.6. This result is known for some classes of the functional operators [1, 8]. One way to prove it is to show that the algebra generated by the corresponding operator Tα contains no non-trivial compact operators – see [1, Theorem 8.3]. On the other hand, the approach used here allows us to obtain certain additional information concerning the kernels of the operators under consideration. Remark 2.7. In theory of dynamical systems, the systems with complete wander- ing sets are called complete dissipative systems [13]. Thus, complete dissipative systems are Fredholm if and only if they are invertible. Acknowledgment The author would like to thank Alexei Karlovich for pointing out reference [6].
- 53. 42 V.D. Didenko References [1] A. Antonevich, Linear Functional Equations. Operator Approach. Operator Theory. Advances and Applications, Vol. 83, Birkhäuser, Basel-Boston-Berlin, 1996. [2] A. Antonevich, A. Lebedev, Functional Differential Equations: C∗ -Theory. Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 70. Longman, Harlow, 1994. [3] N.V. Azbelev, G.G. Islamov, A certain class of functional-differential equations. Dif- ferencial’nye Uravnenija, 12 (1976), 417–427. [4] G. Belitskii, V. Tkachenko, One-dimensional functional equations. Operator Theory: Advances and Applications, Vol. 144, Birkhäuser, Basel, 2003. [5] E. Castillo, A. Iglesias, R. Ruı́z-Cobo, Functional equations in applied sciences. Mathematics in Science and Engineering, Vol. 199, Elsevier, Amsterdam, 2005. [6] A.V. Chistyakov, A pathological counterexample to the non-Fredholmness conjecture in algebras of weighted shift operators, Izv. Vyssh. Uchebn. Zaved. Mat., No. 10 (1995), 76–86. [7] V.D. Didenko, Fredholm properties of multivariate refinement equations. Math. Meth. Appl. Sci., 30 (2007), 1639–1644 [8] Yu.I. Karlovich, V.G. Kravchenko, G.S. Litvinchuk, Invertibility of functional oper- ators in Banach spaces. In: Functional-differential equations, Perm. Politekh. Inst., Perm’, 1990, 18–58. [9] V.G. Kravchenko, G.S. Litvinchuk, Introduction to the theory of singular integral operators with shift. Mathematics and its Applications, Vol. 289, Kluwer, Dordrecht, 1994. [10] M. Kuczma, Functional equations in single variable. Monografie Matematyczne, Vol. 46, PWN, Warszawa, 1968. [11] V.G. Kurbatov, A conjecture in the theory of functional-differential equations, Dif- ferentsial’nye Uravnenija, 14 (1978), 2074–2075. [12] G.S. Litvinchuk, Solvability theory of boundary value problems and singular integral equations with shift. Mathematics and its Applications, Vol. 523, Kluwer, Dordrecht, 2000. [13] P.J. Nicholls, The Ergodic Theory of Discrete Groups. London Mathematical Society Lecture Note Series, Vol. 143, Cambridge University press, Cambridge, 1989. Victor D. Didenko Department of Mathematics University Brunei Darussalam Bandar Seri Begawan BE1410 Brunei e-mail: diviol@gmail.com
- 54. Operator Theory: Advances and Applications, Vol. 210, 43–77 c 2010 Springer Basel AG Lions’ Lemma, Korn’s Inequalities and the Lamé Operator on Hypersurfaces Roland Duduchava Dedicated to my friend and colleague Nikolai Vasilevski on the occasion of his 60th birthday anniversary Abstract. We investigate partial differential equations on hypersurfaces writ- ten in the Cartesian coordinates of the ambient space. In particular, we gen- eralize essentially Lions’ Lemma, prove Korn’s inequality and establish the unique continuation property from the boundary for Killing’s vector fields, which are analogues of rigid motions in the Euclidean space. The obtained results, the Lax-Milgram lemma and some other results are applied to the investigation of the basic Dirichlet and Neumann boundary value problems for the Lamé equation on a hypersurface. Mathematics Subject Classification (2000). 35J57, 74J35, 58J32. Keywords. Lions’s Lemma, Korn’s inequality, Killing’s fields, Lax-Milgram lemma, Lamé equation, Boundary value problems. Introduction Partial differential equations (PDEs) on hypersurfaces and corresponding bound- ary value problems (BVPs) appear rather often in applications: see [Ha1, §72] for the heat conduction by surfaces, [Ar1, §10] for the equations of surface flow, [Ci1], [Ci3],[Ci4], [Ko2], [Go1] for thin flexural shell problems in elasticity, [AC1] for the vacuum Einstein equations describing gravitational fields, [TZ1, TW1] for the Navier-Stokes equations on spherical domains and spheres, [MM1] for minimal surfaces, [AMM1] for diffusion by surfaces, as well as the references therein. Fur- thermore, such equations arise naturally while studying the asymptotic behavior of solutions to elliptic boundary value problems in a neighborhood of conical points (see the classical reference [Ko1]). The investigation was supported by the grant of the Georgian National Science Foundation GNSF/ST07/3-175.
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