![C~nt.. de Recherchee Math~matlqu""
CRM Proceedingl and J ecture Notea
Volume &1. 2010
Canonical de Branges - Rovnyak Model Transfer-Function
Realization for Multivariable Schur-Class Functions
Joseph A. Ball and Vladimir Bolotnikov
ABSTRACT. Associated with any Schur-c111S8 function S(z) (i.e., a contractive
holomorphic function on the the unit disk) is the de Branges-Rovnyak kernel
Ks{~, C) = [1-S{z)S(C)·]/(l- z() and the de Branges- Rovnyak reproducing
kernel Hilbert spsce 1i(Ks). This space plays a prominent role in system
theory as a canonical-model state space for a transfer-function realization of a
given Schur-class function. There has been recent work extending tbe notion
of Schur-cJ1IRS function to several multivariable settings. We bere ma.ke explicit
to what extent the role of de Branges-Rovnyak spaces as the canonical-model
state space for transfer-function realizations of Schur-class functions extends
to multivariable settings.
1. Introduction
Let U and Y be two Hilbert spaces and let L(U, Y) be the space of all bounded
linear operators between U a.nd y. The operator-valued version of the classical
Schur class S(U,Y) is defined to be the set of all holomorphic, contractive L(U, Y)-
valued functions on 1Il. The following equivalent characterizations of the Schur class
are well known. Here we use the notation H2 for the Hardy space over the unit
disk and ~ = H2 ® X for the Hardy space with values in the auxiliary Hilbert
space X.
Theorem 1.1. Let S: 1Il -+ L(U, Y) be given. Then the following are equiva-
lent:
(1) (a) S E S(U, Y), i.e., 8 is holomorphic on 1Il with 118(z)11 ~ 1 for all
z E 1Il.
(b) The opemtor Ms: fez) I-t 8(z)f(z) of multiplication by 8 defines a
contmction opemtor from H~ to H~.
(c) S satisfies the von Neumann inequality: 118(T)1I ~ 1 for any strictly
contmctive opemtor T on a Hilbert space 11., where 8(T) is defined
2000 Mathellwzbcs Subject ClasBifico.tion. 47A57.
Key words and phrasea. Operator-valued functions, Schur multiplier, canonical functional
model. reploducing kernel Hilbert space.
This is the final {olm of the paper.
@20l0 American Mathematical Society
1](https://image.slidesharecdn.com/1124525-250514194345-ce77afbf/85/Hilbert-Spaces-Of-Analytic-Functions-Crm-Proceedings-Lecture-Notes-Thomas-Ransford-18-320.jpg)
![2
(1.1)
J. A. BALL AND V. BOLOTNIKOV
by
00 'Xl
SeT) = L Sn ® rn E C(U ® 1/., Y ® 1/.) if S(z) = LSnz'l.
n=O
(2) The associated kernel
Ks(z, () = Iy - S(z)~«t
1-z(
n=O
is positive on JD) x JD), i.e., there exists an operator-valued functwn H. D-+
C(X, Y) for some auxiliary Hilbert space X so that
(1.2) Ks(z, () = H(z)H«t·
(3) There is an auxiliary Hilbert space X and a umtary conned ng operator
so that S(z) can be expressed as
(1.3) S(z) = D +zO(I - zA)-lB.
(4) S(z) has a realization as in (1.3) where the connect ng opemtorU u
one of (i) isometric, (ii) coisometnc, or (iii controch e.
We note that the equivalence of any of (la), (lb, 1c with 2 and can
be gleaned, e.g., from Lemma V.3.2, Proposition 1.8.3, Proposition V.S.1 and The-
orem V.3.1 in [26]. As for condition (4), it is trivial to see that 3 imphe:s 1
and then it is easy to verify directly that (4) implies (la . Alternatl,-ely, one can
use Lemma 5.1 from Ando's notes [6] to see directly that 4iii implies .fu see
Remark 2.2 below).
The reproducing kernel Hilbert space 1/.(Ks) with the de Branges-Rorn ilk
kernel Ks(z, () is the classical de Branges - Rovnyak reproducmg kernel Hrlberl
space associated with the Schur-class function S which has been much studied O-ef
the years, both as an object in itself and as a tool for other types of applications see
[6,11-13,16-18,20,21,24,27,28,31]). The special role of the de Branges-Ro'Il'8k
space in connection with the transfer-function realization for Schur-class functIons
is illustrated in the following theorem; this form of the results appears at Iea:.
1
implicitly in the work of de Branges Rovnyak [20,21].
Theorem 1.2. Suppose that the function S is in the Schur class S(U, y and e
1-£(Ks) be the associated de Branges Rovnyak space. Define operators A,B,C,D
by
A: fez) 1-+ fez) - f(O) ,
z
C: fez) 1-+ f(O),
B: u I-? S(z) - S(O) u,
z
D: u I-? S(O)u.
Then the operator-block matrix U = l~ ~] has the following properties:
(1) U defines a coisometry from 1/.(Ks) e U to 1/.(Ks) e y.
(2) (C,I) is an observable pair, i.e.,
CAn f ... 0 for all n =0,1,2, ... """'=> J =0 as an element of1-£(Ks).
(3) We recO'Uer S(z) as S(z) = D ;.. zO(I _ .tA) lB.](https://image.slidesharecdn.com/1124525-250514194345-ce77afbf/85/Hilbert-Spaces-Of-Analytic-Functions-Crm-Proceedings-Lecture-Notes-Thomas-Ransford-19-320.jpg)
![4 J. A. BALL AND V. BOLOTNIKOV
Theorem 1.4. Suppose that the function S is in the Schur class S U,y) aM
let K(z,() be the positive kernel on JI} given by (1.5). Define operators A,B,c fi
by
A' rf(z)1 t-+ r[fez) - f(0)1Iz 1
. Lg(z) Lzg(z) - S(z)f(O) ,
B
~ . f([S(z) - S(O 1z u1
. u I-t l (I - S(z)*SOu
8: rf(z)1 t-+ f(O)
Lg(z) ,
D: u I-t S(O)u.
Then the operator-block matrix U = [~ ~1satisfies the following:
(1) Udefines a unitary operator /rom 1£(Ks) G3 U onto 1£ Ks G3 Y·
(2) U is a closely connected operator colligation, i.e.,
V{Ran..4.nB, Ran..4.*nC*} = 1£(Ks).
n~O
(3) We recover S(z) as S(z) = D+zC(I - z..4.)-1B.
(4) If [~: g:] :X G3 U -+ X G3 Y is any other opemtor colligatwn sahsfym9
conditions (1), (2), (3) above (with X in place of1£ Ks , then there IS
unitary operator U: 1£(Ks) -+ X so that
rU
01 fA Bl fA' B'1 ru 01
L
0 Iy lC DJ= L
0' D' lo Iu'
Our goal in this article is to present multivariable analogues of Theorem 1.2.
The multivariable settings which we shall discuss are (1) the unit ball Bel in C' and
the associated Schur class of contractive multipliers between vector-valued Drury
Arveson spaces lI.u(kd) and 11.y (kd), (2) the polydisk with the associated Schurclass
taken to be the class of contractive operator-valued functions on Dei which satlsfy
a von Neumann inequality, and (3) a more general setting where the underlying
domain is characterized via a polynomial-matrix defining function and the Schur
class is defined by the appropriate analogue of the von Neumann inequality. In the'le
multivariable settings, the analogues of Theorem 1.1 have already been set down
at length elsewhere (see [3,15,23] for the ball case, [1,2,14] for the polydisk case,
and [4,5,9] for the case of domains with polynomial-matrix defining function-see
[8] for a survey). Our emphasis here is to make explicit how Theorem 1.2 can
be extended to these multivariable settings. While the reproducing kernel spaces
themselves appear in a straightforward fashion, the canonical model operators on
these spaces are more muddled: in the coisometric case, while the analogues of
the output operator 0 and the feedthrough operator D are tied down, there is no
canonical choice of the analogue of the state operator A and the input operator
B: A and B are required to solve certain types of Gleason problems; we refer to
[25] and [30, Section 6.6] for some perspective on the Gleason problems in general.
The Gleason property can be formulated also in terms of the adjoint operators A'
and B': the actions of the adjoint operators are prescribed on a certain canonically
prescribed proper subspace of the whole state space. From this latter formulation,
one can see that the Gleason problem, although a.t first sight a.ppearing to be rather
complicated, always has solutions. Also, the adjoint of the colligation matrix, rather
than being isometric, is required only to be isometric on a certain subspace of the
whole space X EB y. With these adjustments, Theorem 1.2 goes through in the](https://image.slidesharecdn.com/1124525-250514194345-ce77afbf/85/Hilbert-Spaces-Of-Analytic-Functions-Crm-Proceedings-Lecture-Notes-Thomas-Ransford-21-320.jpg)
![TRANSFER-FUNCTION REALIZATION 5
three settings. Most of these results a.ppear in [10] for the ba.ll case and in more
implicit form in [14] for the polydisk case, although not in the precise formulation
presented here. The parallel results for the third setting are presented here for the
first time. We plan to discuss multivariable analogs of Theorems 1.3 and 1.4 in a.
future publication.
The paper is organized as follows. After the present Introduction, Section 2
lays out the results for the ball case, Section 3 for the polydisk case, and Section 4
for the case of domains with polynomial-matrix defining function. At the end of
Section 4 we indicate how the results of Sections 2 and 3 can be recovered as special
of the general formalism in Section 4.
2. de Branges-Rovnyak kernel associated with a Schur multiplier on
the Drury - Arveson space
A natural extension of the Szego kernel is the Drury - Arveson kernel
1 1
kd(Z,() = =: ( .
1-Z1(1-"'-Zd(d 1- Z,()Cd
The kernel kd Z, () is positive on Bd x Bd where
Bd = {z = (Zl' ... , Zd) E Cd: (z, z) = Zl2 +... + Zd2 < I}
is the unit ball in Cd, and the associated reproducing kernel Hilbert space ll(kd)
is called the Drury-Arveson space. For X any auxiliary Hilbert space, we use
the shorthand notation llX(kd) for the space ll(kd) ® X of vector-valued Drury-
Arveson-space functions. A holomorphic operator-valued function S: Bd -+ LeU, y)
is to be a Drury Arveson space multiplier if the multiplication operator
Ms: / z t--+ S Z fez) defines a bounded operator from llU(kd) to lly(kd)' In
case in addition Ms defines a contraction operator (IMslop 5 1), we say that S
is in the Schur-multiplier class Sd(U, Y). Then the following theorem is the ana.-
logue of Theorem 1.1 for this setting; this result appears in [10,15,23). The alert
reader will notice that there is no analogue of condition (la) in Theorem 1.1 in the
following theorem.
Theorem 2.1. Let S: Bd -+ l.(U, Y) be given. Then the following are equiva-
lent:
2.1)
1) (b S E Sd(U, Y), i.e., the operator Ms of multiplication by 8 defines a
contraction operator from llU(kd) into lly(kd)'
c) S satisfies the von Neumann inequality: 8(T)1 5 1 for any com-
mutative operator d-tuple T = (Tl' ... ,Td) of operators on a Hilbert
space /C such that the operator-block row matrix [Tl . .. Td] defines
a stnct contraction operator from /Cd into /C, where
SeT) = L Sn®-rn E l.(U®ll,Y®ll)
nEZi
if 8(z) = L 8n zn
.
nEZd
+
Here we use the standard multivariable notation:
z'" = Z~1 .,. Z~d and ~ = 'If1 ...T:;d if n = (nl," ., nd) E Zi.
(2) The associated kemel
(2.2)
K ( 1") = 1y - 8(z)8«>*
s z,... 1 _ (z, ()](https://image.slidesharecdn.com/1124525-250514194345-ce77afbf/85/Hilbert-Spaces-Of-Analytic-Functions-Crm-Proceedings-Lecture-Notes-Thomas-Ransford-22-320.jpg)
![6 J. A. BALL AND V. BOLOTNIKOV
is positive on B xB, i.e., there exists an opemtor-valued function H. B.t -+
C(X, Y) for some a'I.IXiliary Hilbert space X so that Ks(z, <) = H(z H <•
(3) There is an a'I.IXiliary Hilbert space X and a unitary connectmg operator
(2.3) u- [~ ~l= l~ ~1' [~1-+l~
so that 8(z) can be expressed as
(2.4) 8(z) =D +0(1 - Zrow(Z)A)-l Zrow(z)B,
where we have set
Zrow(z) = (zlIx ... zd1x).
(4) 8(z) has a realization as in (2.4) where the connecting opemtorU as an
one of (i) isometric, (ii) coisometric, or (iii) contmctwe.
Remark 2.2. Statement (4iii) concerning contractive realizations is not men-
tioned in [15] but is discussed in [10,23]. The approach in [231 is to show that for
8 of the form (2.4) with U = [~ g] contractive the inequality S T) S 1 h ds
for any commutative operator d-tuple T = (Tlo ••• , Td with (Tl ..• Td1 <I,
Le, one verifies (4iii) :::::} (Ie).
The idea of the second approach in [10] is to embed the contraction U ={~g
into a coisometry [~ ~] = (~ gg~] with associated transfer function of the form.
S(z) = (8(z) 81(z)] equal to an extension of S(z) with a larger input space. From
the coisometry property of [~~] one sees that Ks(z,w) = 0 1 -Zrow z A -1 1-
A·Zrow«)-IC·, i.e., S meets condition (2) for the Schur-class Sd U eUhY With
H(z) =C(l - Zrow(z)A)-l. From the equivalence (lb) ~ (2 , it is easy no" to
read off that 8 E Sd(U, y).
A third approach worked out for the classical case but extendable to multr
variable settings appears in Ando's notes [6, Lemma 5.11. Given a contractiw
colligation U = [~g], one can keep the input and output spaces the same but
enlarge the state space to construct a coisometric colligation U = [~~1having
the same transfer function, namely:
r
Qll Q12 0 0
..] n=m'
• 0 0 0 I 0 ...
A= 0 0 0 0 I ,
c= (0 Q21 Q22 0 o ...], D=D
where
Q = [~i~ ~~~] =(I - UU*)1/2.
In this way one gets a direct proof of (4iii) ==> (4ii).
For colligations U of the form (2.3), it turns out that a somewhat weaker notior.
of coisometry is more useful than simply requiring that U be coisometric.](https://image.slidesharecdn.com/1124525-250514194345-ce77afbf/85/Hilbert-Spaces-Of-Analytic-Functions-Crm-Proceedings-Lecture-Notes-Thomas-Ransford-23-320.jpg)
![TRANSFER·FUNCTION REALIZATION 7
Definition 2.3. The operator-block matrix U of the form (2.3) is weakly coiso-
metric if the restriction of U" to the subspace
(2.5)
is isometric.
Du-:= V
(EBcI
lEY
Zrow«)*(l - A* Zrow«)*)-lO"y C Xd
Y Y
It turllS out that the weak-coisometry property of the colligation (2.3) is exactly
what is needed to guarantee the decomposition
(2.6) Ks(z,() = O(l - Zrow(z)A)(l - A"Zrow«)*)-lC*
of the de Branges-Rovnyak kernel Ks associated with S of the form (2.4) (see
Proposition 1.5 in [10]).
2.1. Weakly coisometric canonical functional-model colligations. As
the kernel Ks given by (2.2) is positive on B x B, we can associate a reproducing
kernel Hilbert space 1l(Ks) just as in the classical case, where now the elements
of 1l Ks are holomorphic Y-valued functions on Bd. In the classical case, as we
from Theorem 1.2, there are canonically defined operators A, B, C, D so that
the operator-block matrix U = [~~l is coisometric from ll(Ks) ffiU to ll(Ks) ffi
Y and yields the essentially unique observable, coisometric realization for S E
S U,y. For the present Drury-Arveson space setting, a similar result holds,
but the operators A, B in the colligation matrix U are not completely uniquely
determined. To expJajn the result, we say that the operator A: ll(Ks) -+ 1l(KS)d
s I es the Glea-c;on problem for 1l(Ks) if the identity
d
2.7 / z) - f(0) = L zk(Afh(z) holds for all f E 1l(Ks),
k=l
(Alh (,,)
where we write Af) z) = •
•
•
E 1l(KS)d. We say that the operator B: U-+
(AIM")
1l Ks d solves the 1l(Ks)-Gleason problem for S if the identity
d
2.8 S z)u - S(O)u = LZk(Bu)k(Z) holds for all u E U.
k=l
Solutions of such Gleason problems are easily characterized in terms of adjoint
operators.
Proposition 2.4. The operator A: ll(Ks) -+ 1l(KS)d solves the Gleason prob-
lem for ll(Ks) (2.7) if and only if A*: 1l(KS)d -+ ll(Ks) has the following action
all. special kernel functions:
2.9) A*: Zrow«)*Ksh ()y ~ Ks(o, ()y - Ksh O)y for all (E Bd
, Y E y.
The operator B: U -+ 1l(KS)d solves the ll(Ks)-Gleason problem for S (2.8) if
and only if B*: 1l(KS)d -+ U has the following action on special kernel functions:
(2.10) B*: Zrow«)*Ks("()Y t-+ S«)*y - S(O)"y for all (E Bd
, Y E y.](https://image.slidesharecdn.com/1124525-250514194345-ce77afbf/85/Hilbert-Spaces-Of-Analytic-Functions-Crm-Proceedings-Lecture-Notes-Thomas-Ransford-24-320.jpg)
![TRANSFER-FUNCTION REALIZATION 11
respectively. The two latter equalities tell us that U· is isometric on the 'Du.
(see (2.5» and therefore U is weakly coisometric. For the proof of part (4) we refer
to [10, Theorem 3.4]. 0
Definition 2.5 does not require U to be a realization for S: representation
(2.4) is automatic once the operators A, B, 0 and D are of the required form.
We can look at this from a different point of view as follows. Let us say that
A: 1I.(Ks) -+ 1I.(KS)d is a contractive solution of the Gleason problem (2.7) if in
addition to (2.7), inequality
d
(2.23) LI(Af)kl~(Ks) ::; Ifll~(Ks) -lIf(O)II~
k=l
holds for every f E 1i(Ks). It is readily seen tha.t inequality (2.23) can be equiva.-
lently written in operator form as
where the operator 0: 1i(Ks) -+ Y is given in (2.12). It therefore follows from
Definition 2 5 that for every canonical functional-model colligation U = l~ g] for
S, the operator A is a contractive solution of the Gleason problem (2.7). The
following theorem provides a converse to this statement.
Theorem 2.10. Let S E Sd(U,Y) be given and let us assume that 0, Dare
!I' en by fonllulas (2.12). Then
1 For every contractive solution A of the Gleason problem (2.7) for 11.(Ks) ,
there exists an operator B : U -+ 11.(Ks) such that U = [~ g] is contrac-
h e and S zs reahzed as in (2.4).
2 E ery such B solves the 1I.(Ks)-Gleason problem (2.8) so that U is a
canonical functional-model colligation.
PROOF. Smce A solves the Gleason problem (2.7) and since 0 is defined as in
2.12 , we conclude as in the proof of Theorem 2.9 that identity (2.17) holds which
is equivalent to (2.19). On account of (2.19), it is readily seen that (2.18) and (2.20)
are equivalent. But (2.20) is just the adjoint form of (2.4) whereas (2.18) coincides
with 2.10 since D = S(O)) which in turn, is equivalent to (2.8) by Proposition 2.4.
Thus, it remains to show that there exists an operator B*: 1I.(Ks) -+ U completely
detew.ined on the subspace 'D C 1I.(Ks) by formula (2.10) and such that U· =
[~: ~: 1is contractive. This demonstration can be found in [10, Theorem 2.4]. 0
3. de Branges - Rovnyak kernels associated with a Schur - Agler-class
function on the polydisk
Here we introduce a generalized Schur class, called Schur - Agler class, associ-
ated with the unit polydisk
Jl)d = {z = (Zl, ••• ,Zd) E Cd: Iz,.1 < 1 for k = 1, ... ,d}.
We define the Schur-Agler class SAd(U,Y) to consist of holomorphic functions
S: nd -+ £(U, Y) such that IS(T) II ::; 1 for any collection of d commuting opera.tors
T = (Tlo"" Td) on a Hilbert space JC with liT,.II < 1 for each k = 1, ... , d where
the operator SeT) is defined 88 in (2.1).](https://image.slidesharecdn.com/1124525-250514194345-ce77afbf/85/Hilbert-Spaces-Of-Analytic-Functions-Crm-Proceedings-Lecture-Notes-Thomas-Ransford-28-320.jpg)
![12 J. A. BALL AND V. BOLOTNIKOV
The following result appears in [1,2,14] and is another multivariable analogup
of Theorem 1.1. The reader will notice that analogues of both (la) and lb fr m
Theorem 1.1 are missing in this theorem.
Theorem 3.1. Let S be a c'(U, Y) -valued function defined on Del. The follow-
ing statements are equivalent:
(3.1)
(1) (c) S belongs to the class SAd(U, Y), i.e., S satisfies the von NeufTl.4Tlrt
inequality IS(Tl. ... ,Td)1I :5 1 for any commutative d-tuple T :::
(T1, ••• , Td) of strict contraction operators on an a1.lXtl,ary Ht1hert
space /C.
(2) There exist positive kernels KI, ... ,Kd:]Dd x]Dd -+ c'(Y) such that/or
every z =(Zl. ... , Zd) and ( = «(1. •.• , (d) in]Dd,
d
ly - S(z)S«()* = 2)1- z.(.)K.(z,().
k=l
(3) There exist Hilbert spaces Xl,"" Xd and a unitary connectmg operator
U of the structured form
[
A B1 lA~l ... A~d ~11l~11 l~11
(3.2) U = 0 D = A:dl A:
dd
~d : ;d -+ ;d
(3.3)
(3.4)
0 1 Od D U Y
so that S(z) can be realized in the form
S(z) = D +0(1 - Zdiag(Z)A) -1 Zdiag(z)B for all z E Dd
where we have set
(4) There exist Hilbert spaces Xl, ... I Xd and a contractive connecnng operator
U of the form (3.2) so that S(z) can be realized in the form (3.3)
Remark 3.2. Although statement (4) in Theorem 3.1 concerning contracti1l
realizations does not appear in [1,2,14], its equivalence to statements (1)-(3) can
be seen by anyone of the three approaches mentioned in Remark 2.2.
Similar to the notion introduced above for the unit-ball case, there is a notion
of weak coisometry for the polydisk setting as follows.
Definition 3.3. The operator-block matrix U of the form (3.2) is weakly cois/)"
metric if the restriction of U· to the subspace
(3.5) Vu.:= V [Zdlag«()*(1 - A~Zdiag«()·)-lO*Y] c [~d]
<eDd
IIeY
is isometric.](https://image.slidesharecdn.com/1124525-250514194345-ce77afbf/85/Hilbert-Spaces-Of-Analytic-Functions-Crm-Proceedings-Lecture-Notes-Thomas-Ransford-29-320.jpg)
![14 J. A. BALL AND V. BOLOTNIKOV
It is clear that kers ={J E EB~=lll(Ki) : /l(z) +... + Jd(Z) == O}. If we let
l
Kl(Z,()]
11'(z, () ;= : '
Kd(z,()
(3.10)
we observe that by the reproducing kernel property,
d
(3.11) (I, 11'(., ()Y)E9t~ll£(K,) = L)Ii,K i (·, ()Y}l£(K,)
i=1
so that
(3.12) s·: 1K(·, ()y --+ 11'(., ()y.
Furthermore,
d
(kers).L = V11'(.,()y c ffill(Kk ).
(EDd k=1
yE)I
We next introduce the subspace
(3.13) v = VZdiag«)"''ll'(·,()y
(ED"
yE)I
of EB~=11l(Kk) and observe that its orthogonal complement can be described as
d d d
V.L = {I = ~Ii E ~ll(K,) : ~Z.i,(z) =o}-
In addition, the straightforward computation
d
1111'(·,()YIla,~=l1t(Kk) = 2:(Kk«,()Y,Y}Y = (K«,()y,y}y = 1I1K(·,()y ~(K)
k=1
combined with (3.12) shows that s* is an isometry, Le., that s is a coisometry· We
remark that all the items introduced so far are uniquely determined from decom-
position (3.1).
Given an operator-block matrix A = [A'Jlt..,=1 acting on EB~=1 ll(K,), we;:}
that A solves the structured Gleason problem lor the kernel collection {KlI""
if the identity
d
(3.14) !I(z) +... +Id(Z) - [/1(0) + ... + ideO)] = :Lz.(Af),(z)
,=1
d d md (Af).(Z)
holds for all 1= EB.=1 I. E EBi =1 ll(Ki), where we write (AI)(z) == 1;17>==1
E EB~_lll(K.). Note that (3.14) can be written more compactly as
d d
(3.15) (sl)(z) - (sl)(O) = Lz,(A••f)(z) for all I E E9ll(K/r)'
, 1 " 1](https://image.slidesharecdn.com/1124525-250514194345-ce77afbf/85/Hilbert-Spaces-Of-Analytic-Functions-Crm-Proceedings-Lecture-Notes-Thomas-Ransford-31-320.jpg)
![16 J. A. BALL AND V. BOLOTNIKOV
The following definition of a canonical functional-model colligation is the ~
logue of Definition 2.5 for the polydisk setting.
Definition 3.5. Given S E SAd(U,Y), we shall say that the block-()peratr.t
matrix U = [~g1of the form (3.2) is a canonical functional-model coli gatlon
associated with the Agler decomposition (3.1) for S if
(1) U is contractive and the state space equals ffi~=11l(K.).
(2) A: EB1=11l(Ki) -4 ffi1=11l(Ki) solves the structured Gleason problem
(3.15).
(3) B: U -4 EB1=11l(Ki) solves the structured Gleason problem 3.18 for S
(4) The operators C: EBt=11l(Ki) ~ Y and D: U ~ Y are given by
(3.21) C: f(z) ~ (81)(0), D: 11. ~ S(O)1I..
Remark 3.6. For C and D defined in (3.21), the adjoint operators are gi."1!Ii
by
(3.22) C*: Y~ 'll'(',O)y D*: y ~ S(O)*y.
The formula for D* is obvious while the formula for C* follows from equalities
(I, C*Y)(Bt=l1i(K;) = (Cf, y)y = ((sl)(O), y}y = (f,'ll' ·,O)y (B =11l K
holding for every f E EB1=11l(Ki).
Theorem 3.7. Let S be a given function in the Schur-Agler class SA.:! U,y)
and suppose that we are given an Agler decomposition (3.1) fOT S. Then there e:nsts
a canonical functional-model colligation associated with {K1, .. " Kd}.
PROOF. Let us represent a given Agler decomposition (3.1) in the inner product
form as
d
l)(iKi(" ()y, ZiKi(', z)y')1i(K,) + (y, y')y
~l d
= L (Ki (·, ()y, K.(., z)y'ht(K,> + (S«()*y, S(z)'!I~u,
i=1
or equivalently, as
(3.23) ([Zdiag(();'ll'("()y], [Zdiag(z);!(.,z)y'])
(EB~=l 'H(K,))GlY
1['ll'("()Y) ['ll'("Z)y'])
= S«()*y , S(z)*y' «(B~=l1l(K,))$U
where l' is given in (3.10). The latter identity implies that the map
(3.24) V: [Zdias(()*'ll'("()y] ~ ['ll'(" ()y]
y S«()*y
extends by linearity and continuity to an isometry from Vv =V EB Y (8. subspace
of (EBt=lll(Ki )) EB Y-(3.13) for definition of V) onto
'R- = V ['ll'("()Y] C [ffi~_lll(Ki)]
V S«()*y U '
~EDd,IlEY](https://image.slidesharecdn.com/1124525-250514194345-ce77afbf/85/Hilbert-Spaces-Of-Analytic-Functions-Crm-Proceedings-Lecture-Notes-Thomas-Ransford-33-320.jpg)
![TRANSFER-FUNCTION REALIZATION
-
Let us extend V to a. contraction U· :
(3.25)
Computing the top and bottom components in (3.25) gives
(3.26)
(3.27)
A·Zdie.g«).'11'(., ()y +C·y = '11'(" ()y,
B· Zdie.g«)·'11'(·, ()y + D·y = S«ry.
11
Letting ( = 0 in the latter equalities yields (3.22) which means that C and D are
ofthe requisite form (3.21). By substituting (3.22) into (3.26) and (3.27), we arrive
at (3.19) and (3.20) which in turn are equivalent to (3.15) and (3.18), respectively.
Thus, U meets all the requirements of Definition 3.5. 0
We have the following parallel of Remark 2.8 for the polydisk setting.
Remark 3.B. As a consequence of the isometric property of the operator
-
V 3.24) introduced in the proof of Theorem 3.7, formulas (3.19) and (3.20) can
be extended. by linearity and continuity to define uniquely determined operators
AD: 'D -+ ffi~=lll(K.) and BD: 'D -+ U where the subspace 'D of ffi~=lll(Ki) is
defined. in (3.13). In view of Proposition 3.4, we see that the existence question
is then settled.: any opemtor A: ffi~=l1-£(Ki) -+ ffi~=lll(Ki) such that A· is an
ofAD from 'D to all offfi~=lll(Ki) is a solution of the structured Gleason
3.15) and any opemtor B: U -+ ffi1=11-£(Ki ) so that B· is an extension
of the operator BD: 'D -+ U is a solution of the structured Gleason problem (3.18)
forS.
In the polydisk setting we use the following definition of observability: given
an operator A on ffi~=l.:ti and an operator C: ffi~=lXi -+ y, the pair (C,A)
wi11 be called obsenJable if equalities C(1 - Zdiag(z)A)-lPx,x = 0 for all z in a
neighborhood of the origin and for all i = 1, ... , d forces x = 0 in ffi~=l Xi. The
latter is equivalent to the equality
3.28 V Px.(1 - A·Zdiag(Z)·)-lC·y = Xi for i = 1, ... , d
zEa,I/E)/
for some neighborhood f:l. of the origin in Cd. The following theorem is the analogue
of Theorem 1.2 for the polydisk setting; portions of this theorem appear already in
{14, Section 3.3.1].
Theorem 3.9. Let S be a function in the Schur - Agler class SAd(U, Y) with
a gtven Agler decomposition {K1 , •••• Kd} for S and let us suppose that
(3.29)
u a canonical functional-model colligation associated with this decomposition. Then:
(I) U is weakly coisometric.
(2) The pair (C. A) is obsenJable in the sense of (3.28).
(3) We recover S(z) as S(z) = D +0(1 - Zdiag(z)A)-lZdle.g(z)B.](https://image.slidesharecdn.com/1124525-250514194345-ce77afbf/85/Hilbert-Spaces-Of-Analytic-Functions-Crm-Proceedings-Lecture-Notes-Thomas-Ransford-34-320.jpg)
![TRANSFER-FUNCTION REALIZATION 21
Step 1. Let A of the form (3.17) solve the Gleason problem (3.15). Then
d
(3.40) 0(1 - Zdia.g(z)A)-1f = (81)(Z) Z E Jl)dj f E E91i(Ki)
,=1
where sand C are defined in (3.9) and (3.21), respectively.
PROOF OF STEP 1. To show that identity (3.15) is equivalent to (3.40) we take
A in the form (3.17) and define the operators
Ale 0 0
0 A2• •
~
• • •
(3.41) Ale = A2e = Ad. = • ,
• , • , ...,
• • 0
• •
0 0 Ad.
d
SO that A•• : E9~=1 1i(K,) -+ E9~=11i(Ki) and A = 2:Ai.' On account of (3.41)
.=1
and due to the block structure (3.4) of Zdia.g(Z) we have
-1 .... ,. -1
(1 - Zdia.g(z)A) = (1 - ZlAh - '" - ZdAd.)
Applying the operator e(l - Zdiag(z)A)-1 to an arbitrary f E E9~=11i(Ki) and
making use of formula (3.21) for C, we get
3.42 e 1 - Zdtag(z)A)-lf
00
= e~)zlAh +... + zdAd.)kf
k=O
d d
=(81)(0) +LZi(sAi.I)(O) + 2: zizj(sAi.Aj.I)(O) +....
•=1 i,j=l
On the other ba,nd, by writing (3.15) in the form
d
(sl)(z) = (sl)(O) +L:zi(sAi.I)(Z)
;'=1
and iterating the latter fOlmula for each f E E9~=11i(Ki)' we get
(3.43) (s/)(z)
d d
= (81)(0) + L Z;1 (SA31.1)(0) + L Zh (sAh.Ail.f)(O) +.,.
]1=1 h=l
d
+ L Zjh [(sAj/o. '" A3~.Ajl.I)(0) +...J... .
310=1](https://image.slidesharecdn.com/1124525-250514194345-ce77afbf/85/Hilbert-Spaces-Of-Analytic-Functions-Crm-Proceedings-Lecture-Notes-Thomas-Ransford-38-320.jpg)
![22 J. A. BALL AND V. BOLOTNlKOV
Since the right-hand side expressions in (3.42) and (3.43) are identical, 3.40 f0l-
lows. Now we have from (3.40)
(I, (1 - A*Zdlag(Z)*)-lC*y) =(C(1 - Zdiag(Z)A)-1f, y) = «sf) z ,y
= /,T·,zy
for every z E lDid
and f E EBt=l'H.(Ki ), and thus, equality (3.31) holds. 0
Step 2. Given operators A, C and D with A 0/ the form 3.17) equal to
contractive solution of the structured Gleason problem for the kernel collechon
{Kl, ... ,Kd} and with C and D given by (3.21), ifU = (~Zl t.s a ~e
realization of S for some operator B =EB B,: U ~ ffi~=1 'H.(K.), then B salta jh,
EB~=l1i(Kk)-Gleason problem for S, i.e., B satisfies identity 3.18.
PROOF OF STEP 2. Since U = [~Z] is a realization for S, equalrty 3.33
holds. Making use of equality (3.31) (which holds by Step lone can write 3.33
as
B*Zdiag«)*'lI.'(·, ()y + D*y = S ( .y
or, in view of formula (3.21) for D, as
(3.44) B* Zdiag«r'lI.'(·, ()y = S«)*y - S 0 .y.
Taking the inner product of both parts in (3.44) with an arbitrary funw n JE
EBt=l1i(Ki) leads us to Zdiag(z)Bu = S(z)u-S(O u which is the same as 3.18 0
To complete the proof of the theorem, it suffices to show that there exists an
operator B : U -+ EBt=l 1i(Ki) such that equality (3.33 holds for every u EU and
the operator matrix
(3.45) U* = [A* C*1. rffit=1 'H.(K·)1 ~ rffi~=1 'H. K )1
B* D* . l y l u
is a contraction. As we have seen, equality (3.33) is equivalent to (3.44 , v-htch
in turn, defines B* on the space 'D introduced in (3.13). Let us define B: 1> -i
EB:=l1i(Ki ) by the formula
B: Z«)*'lI.'(.,()y = S«)*y - S(O)·y
and subsequent extension by linearity and continuity; it is a consequence of the
isometric property of the operator V in (3.24) that the extension is well-defined
and bounded. We arrive at the following contractive matrix-completion problero:
d -
find B: U -+ EBi=l 1l(K,) such that B*!'D = B and such that U· of the form (3.45
is a contraction. Following [10] we convert this problem to a standard matrix-
completion problem as follows. Define operators
d d
Tn: Vi -+ E91i(K,), T12: 'D ~y ~ E9ll(K,),
.-1 =1
by
(3.46)](https://image.slidesharecdn.com/1124525-250514194345-ce77afbf/85/Hilbert-Spaces-Of-Analytic-Functions-Crm-Proceedings-Lecture-Notes-Thomas-Ransford-39-320.jpg)
![TRANSFER-FUNCTION REALIZATION 23
Identifying we then can represent U· from (3.45) as
(3.47) 1>.1. -+ EB~=l 1i(K,)
1>Ef)Y U
where X = B*v.L is unknown. Thus, an operator B gives rise to a canonical
functional-model realization U = t~ g1of S if and only if it is of the form
X. 'D.1. It
B = B* :U -+ 'D ~ E!11i(K,)
,=1
where X is any solution of the contractive matrix-completion problem (3.47). But
this is a standard matrix-completion problem which can be handled by the well-
known Parrott's result [291: it has a solution X if and only if the obvious necessary
conditions hold:
3.48
:Making use of the definitions of Tn. T12,T22 from (3.46), we get more explicitly
Tt2 A"v C"
[Tu T121 = [A* Co01. T22 = B D'"
Thus the first in (3.48) is contractive since A is a contractive solution
of the stru tured Gleason problem (3.15), while the second expression collapses to
-
V see f rmula 3.24) which is isometric by (3.1). We conclude that the necessary
nditi ns 3.48 are satisfied and hence, by the result of [29], there exists a solution
X to problem 3.47. This completes the proof of the theorem. 0
4. de Branges - Rovnyak kernels associated with a Schur- Agler-class
function on a domain with matrix-polynomial defining function
A generalized Schur class containing all those discussed in the previous sections
as special was introduced and studied in [4,9] (see also [5] for the scalar-valued
and can be defined as follows. Let Q be a p x q matrix-valued polynomial
4.1
such that
4.2
Q(z) =
qu(z) .. . q1q(Z)
•
,
,
•
•
•
Q(O) = 0
and let 'DQ E en be the domain defined by
'DQ = {z E en: Q(z)1I < I}.
Now we recall the Schur Agler class SAQ(U, Y) that consists, by definition, of
£. U, Y)-valued functions S(z) = S(Zl" '" zn) analytic on 'DQ and such that IIS(T) II
::; 1 for any collection of n commuting operators T = (T1, ... ,Tn) on a. Hilbert
space K:, subject to Q(T)U < 1. By [5, Lemma 1], the Taylor joint spectrum of
the commuting n-tuple T = (T1,.•. ,Tn) is contained in 'DQ whenever IIQ(T)II < 1,
and hence SeT) is well defined by the Taylor functional calculus (see [19]) for any
£. U, Y)-valued function S which is analytic on 'DQ. Upon using K: = e and T, = z,](https://image.slidesharecdn.com/1124525-250514194345-ce77afbf/85/Hilbert-Spaces-Of-Analytic-Functions-Crm-Proceedings-Lecture-Notes-Thomas-Ransford-40-320.jpg)
![J. A. BALL AND V BOLOTNIKOV
for j 1, ... ,n where (Zh"" zn) is a point in VQ we conclude that any fun<'t
SAQ(U,Y) is contractive-valued, and thus, the class SAQ U,Y) is th Ia
the Schur class SPQ (U, Y) of contractive valued functions analyt'c on 'DQ B
von Neumann result, in the case when Q(z) =z, these classes coinClde- in getll!r:!l.
SAQ(U, Y) is a proper subclass of SPQ(U, Y). The following result appears
(see also [5] for the scalar-valued case U - Y = C) and is yet another m b
analogue of Theorem 1.1. We will often abuse notation and will wnte Q z lIlStead
of Q(z) ® 1 where 1 is the identity operator on an appropnate Hilbert space
from the context. When the following theorem is viewed as a parallel fTh rem
we see that, just as in the polydisk setting, there is no parallel to rut lIS
and (lb).
Theorem 4.1. Let S be a £(U, Y)-valued fund on defined on'DQ The
lowing statements are equivalent:
(4.3)
(1) (c) S belongs to SAQ(U,y),
(2) There exists a positive kernel
(
1K~1 .. . lK~p1
IK =: : : 'DQ
1Kpl IKpp
which provides a Q-Agler decomposition f r S .e, su that f
z,( E'DQ,
P q
(4.4) Iy - S(z)S«()* = L IKkk (Z, () - L L q z QT~ z (
k=1 k-l,=1
(2') There exist an auxiliary Hilbert space X a d a fu t&
(4.5) H(z) = [Hl(Z) '" Hp(z]
analytic on'DQ with values In .c(XP,)7) so that f revery z, ( E 'DQ
(4.6) Iy - S(z)S«()* =H(z)(Ix - Q z Q ( * H ( *.
(4.7)
(3) There exist an auxiliary Hilbert space X and a u ~ tary ronnec g perot
U of the form
u = [~ ~]: [~] ~ [~]
so that S(z) can be realized in the form
(4.8) S(z) - D +C(Ixp - Q(z)A) lQ(z)B for all Z E'DQ.
(4) There exist an auxiliary Hilbert space X and a contractive connecun9 r
erator U of the form (4.7) so that S(z) can be realized in the form (4.
Remark 4.2. If S = [~!~ ~~~] E SAQ(U1$U2,)71 $Y2), then the block~t~
5'J
belongs to the Schur Agler class SAQ(U"YJ ) for i,j = 1,2. For the pro '
suffices to note that IIS'J(T)II $ IIS(T).](https://image.slidesharecdn.com/1124525-250514194345-ce77afbf/85/Hilbert-Spaces-Of-Analytic-Functions-Crm-Proceedings-Lecture-Notes-Thomas-Ransford-41-320.jpg)
![~6 J. A. BALL AND V. BOLOTNIKOV
decomposition
(4.12)
l
lK(I) (Z, ()
lK(z,() =
o
where lK(a) in turn has the form
l
lK(a)
11
lK(a) = :
lK(a)
p",l
Under the normalizing assumption that lK has this block diagonal f rm -1.12
Q(z) is written as a direct sum of irreducible pieces (4.11 ,the nstruct1 us to
follow can be done with more efficient labeling but at the cost of an additI nal 1lf
of notation. We therefore shall assume in the sequel that this diag nal structure •
not been taken into account (or that the matrix polynomial Q is already irredu e
until the very end of the paper where we explain how the polydisk settIDa can
seen as an instance of the general setting.
As in the previous particular settings of the ball and of the polydIsk, we mtro-
duce the weak-coisometry property as the property equivalent to 4.10 llapsmo
to
1 - S(z)S«)* = e(1 - Q(z)A)-I(I - Q(z)Q (* 1- A*Q ( • -Ie-.
Definition 4.5. The operator-block matrix U of the form 4.7 is weakl lSO-
metric if the restriction of U" to the subspace
(4.13) .- V lQ«)"(1 - A*Q«)* -lc·y1 (Xq
1
'Du··- c y
(EVQ Y
yEY
is isometric.
Due to assumption (4.2), the space 'Du. splits in the form 'Du. ='DEBYwhere
(4.14) 'D = V Q«)*(I - A*Q«)*)-IC"y c Xq
•
(EVQ,yEY
4.1. Weakly coisometric canonical functional-model Q-realizations.
Let us suppose that we are given a function S in the Schur Agler class SAQ u,Y
together with an Agler decomposition lK. as in (4.3) (so (4.4) is satisfied). We'9l,u
use the notation Q.k«) for the k-th column of the polynomial matrix Q. What
actually comes up often is the transpose:
(4.15) Q.k«)T = (qlk«) q2k«) ... ~k«)]'
Note that with this notation the Q-Agler decomposition for S (4.4) can be written
more compactly as
p q
(4.16) 1:v - S(z)S«)* =L KIt,k(Z, () - L QJ)(z)lK(z, ()QJJ«)"
It I j 1
an expression more suggestive of the Agler decomposition (3.1) for the pol)~
case.](https://image.slidesharecdn.com/1124525-250514194345-ce77afbf/85/Hilbert-Spaces-Of-Analytic-Functions-Crm-Proceedings-Lecture-Notes-Thomas-Ransford-43-320.jpg)
![28 J. A. BALL AND V. BOLOTNIKOV
(4.1), (4.3), (4.15) and (4.20). Letting
(4.23) 1l'(z, () := :
llK(Z' ~)E1J
lK(z, ()Ep
for short, we then can write formulas (4.19), (4.21) more compactly as
(4.24) A*Q«)*1l'(·, ()y =1l'(', ()y -1l'(" O)y,
(4.25) B*Q«)"1l'(·, ()y =8«)*y - 8(0)*y
where now Q«)* is to be interpreted as Q«)" ® 11£(lK)' In the following computa-
tions, Q«)* is either Q«)" ® Iy or Q«)" ® 11£(K) according to the context. B)
the reproducing kernel property, we have for every f = EI):=1 I" E 1llK. P,
p p
(4.26) (I,'lr(',()Y)l£(IK)P = LU",lK(., ()E"y)l£(K) = I: Ei.f,,« ,y y
"=1 "=1
Therefore,
(4.27) (I,1l'(·,()y-1l'(·,O)Y)l£(IK)P = <t;u",,,«)-f,,,,, 0 ),Y)y'
On the other hand, it follows again from (4.26) that
(I,A*Q(z)*1l'("Z)Y)l£(IK)P = (Q(z)AI,1l'("Z)Y)l£(K)P = <t,[Q(Z)AllJJ Z,Y y
and since
p p q
(4.28) ~[Q(z)Alkj(z) = L L <l.i"(z)[AJ],,,](z)
j=1 j=I,,=1
we get
(I, A*Q(z)"1l'(·, z)y)1£(lK)p = <
t Q.,,(z)T[Afl,,(z), Y) y'
"=1 Y the
Since the last equality and (4.27) hold for every f E 1l(K)P, ( E VQ and y ~val~ce
equivalence of (4.17) and (4.24) (which is the same as (4.19)) folloWS. EqUl
of (4.18) and (4.25) follows by the same argument from equalities
(1£, S«)"y - S(O)"y)u = (8«)1£ - 8(0)1£, y}y
and
(1£, B*Q(z)*1l'(·, z)y)u =(Q(z)Bu, 1l'(', Z)Y)l£(IK)P
=(t[Q(z) Bulj,j (z),y) = (:tQ.,(z)T[Bti1.(M),
3=1 y "==1 Cl
holding for all 1£ E U and y E :V.](https://image.slidesharecdn.com/1124525-250514194345-ce77afbf/85/Hilbert-Spaces-Of-Analytic-Functions-Crm-Proceedings-Lecture-Notes-Thomas-Ransford-45-320.jpg)
![30 J. A. BALL AND V. BOLOTNIKOV
The latter can be written in the matrix form as
(4.32) / rQ«()*1l'(·, ()Y1 ' rQ(Z)*1l',(.,Z)Y'1)
L Y L Y 'H(K)geY
( fT ("( y1 fT ',Z y'1
= ls (ty ,lS Z *y'
The latter identity implies that the formula
(4.33) V: rQ«)*1l'("()Y1 r-+ rT "()y1
L y lS ( .y
extends by continuity to define the isometry from 1 = V ffi Y c 'Hex q Ee Y see
(4.14) for definition of 'D) onto
nv = V l~~(f:~1
c l1l~)p1·
(E'DQ
lIEY
Let us extend V to a contraction U*: ['Hc;)9 1~ ['H~)P]. Thus,
(4.34) U* - [A* c*1· fQ«)*T(., ()y1 r-+ fT("()Y1
- B* D* . l y lS«)*y •
Computation of the top and bottom components in (3.25) gives
(4.35) A*Q«)*1l'(., ()y +C*y = T(·, ()y,
(4.36) B*Q«)*T(·, ()y + D*y = S«)*y.
Letting ( =0 in the latter equalities and taking into account (4.2) leads ~tOt~4:Oof
from which we see that C and D are of the requisite form (4.29). Substl
tu
10uiv-
(4.30) into (4.35) and (4.36) then leads us to (4.24) and (4.25) w~ch a:n:ucal
alent to (4.17) and (4.18), respectively. Thus we conclude that U 18 a 0
functional-model colligation as wanted.
. an operatol'
For this general setting we define observability as follows: gI~n called Q.
A: XI' -+ xq and an operator C: XI' ~ Y, the pair (C, A) will. be
b
hOod of
observable if the identities C(I - Q(Z)A)-II.;x = 0 for all z in a neigh or denote
the origin and for all i = 1, ... ,p forces x = 0 in X. By Z. : X -+ X" we](https://image.slidesharecdn.com/1124525-250514194345-ce77afbf/85/Hilbert-Spaces-Of-Analytic-Functions-Crm-Proceedings-Lecture-Notes-Thomas-Ransford-47-320.jpg)
![TRANSFER-FUNCTION REALIZATION a1
the inclusion ma.p which embeds X into the i-th component of XI' = X ffi .•. ~ X.
Thus
0 Xl
• •
• •
• •
(4.37) ~: x,t-+ X, and r·
• • X, t-+ X•.
• •
• •
• •
0 xI'
The Q-observability can be equivalently defined in terms of adjoint operators as
(4.38 VrI;(I A*Q(z)*) lC*y:zEb.,yEY,i=1, ... ,p}=X
where b. is some neighborhood of the origin in en. The following theorem is the
analogue of Theorem 1.2 for the present general setting.
Theorem 4.9. Let S be a function in the Schur-Agler class SAQ(U, Y), let
the poSuve kernellK. of the foun (4.3) provide an Agler decomposition (4.4) for S
a d suppose that U = t~ Z) :ll(lK.)p ffi U -t ll(lK.)q ffi Y is a canonical functional-
model lltgatton assoctated with 8 and lK.. Then the following hold:
1 U 1.8 eakly C01.8ometnc.
2 The p (C,A) 1.8 Q-observable in the sense of (4.38).
439
3 "' reco er S as 8(z) = D +C(I - Q(z)A)-lQ(z)B.
4 If U = ~ ~ : XP ffi U -t xq ffi Y is another colligation matrix enjoying
p ert es (1), (2), (3) above, then there is a canonical functional-model
-
col 'g tum U for (8, lK.) such that U and U are unitarily equivalent in the
sense that there is a unitary operator U: X -t 1l(lK.) so that
A B
C D
EI1~=l U
o
o _ EI1?=l U 0
Iy - 0 Iu
- -
A B
CD'
PROOF. Let U = t~ Z) be a. canonical functional-model realization of 8 asso-
ciated with a fixed Agler decomposition (4.4). Then combining equalities (4.24),
425 equivalent to the given (4.17) and (4.18) by Proposition 4.6) and also for-
mulas 4.30 equivalent to the given (4.29)) gives
4.40 1'("( y = (I - A*Q(()*)-l']['(., O)y = (I - A*Q(()*)-lC*y
and
4.41 S(()*y = 8(0)*y +B*Q(()*']['(', ()y = D*y +B*Q(()*,][,(., ()y.
Substituting (4.40) into (4.41) and taking into account that y E Y is arbitrary, we
get
4.42) 8(()* = S(O)* + B*Q(()*(I - A*Q(()*)-lC·
which proves part (3) of the theorem. Also we have from (4.40)
V r.(I - A*Q(()*)-lC·y = V :r;']['(.,()y
C;E"DQ IIEY.
t-l,....,p
CE"DQ,IIEY,
t 1,..'IP](https://image.slidesharecdn.com/1124525-250514194345-ce77afbf/85/Hilbert-Spaces-Of-Analytic-Functions-Crm-Proceedings-Lecture-Notes-Thomas-Ransford-48-320.jpg)
![32 J. A. BALL AND V BOLOTNIKOV
and we can proceed due to (4.37) and (4.23) as follows:
V X;1l'(.,()y= V JK(·,()E.V= V K(',()y=1£X
C;EDQ,lIEY,
i-I,...,,,
C;E'DQ,lIEY,
1.=1,... ,1'
<E'DQ
yEY
Thus the pair (C, A) is Q-observable in the sense of (4.38). On the other ban
equalities (4.24), (4.25) are equivalent to (4.34). Substituting (4.40 into 4.34 an
into identity (4.32) (for z =( and V =V') gives
• [Q()'(I - A*Q()*)-IC*v1_ r(1- A*Q ( • -1C*V}
u y - l S <'y
and
respectively. The two latter equalities show that U· is isometric on the space Vu
(see (4.13)) and therefore U is weakly coisometric.
To prove part (4), let us assume that
(4.43) 8(z) = 8(0) +O(I - Q(z)A)-IQ(z B
is a weakly coisometric realization of 8 with the state space X and such that the
pair (0, A) is Q-observable in the sense of (4.38). Then
1- S(z)S()* = O(I - Q(z)A)-l(I - Q(z)Q (t 1- A*Q ( * -Ie-
which means that 8 admits a representation (4.6) with H z = 0 1- Q z A-
Let I; be given as in (4.37). Representing H in the form 4.5 with
(4.44)
we then conclude from Remark 4.3 that 8 admits the Agler decompositi n 44
with
for i,j = 1, ... ,p.
Let 1£(IK) be the reproducing kernel Hilbert space associated with the posithe
kernellK =[1K'31~,j=I' Let us arrange the functions (4.44) as follows
(4.45)
(
HI.(Z)] lo(I-Q(.Z)A)_lXl]
G(z) := : = : .
Hp(z) O(I - Q(z)A)-IIp
Since by construction lK(z,() = G(z)G()* and since (0,...1) is Q-observable, the
formula
(4.46) U: x ~ G(z)x
defines a unitary map from X onto 1£(IK). Let us define the operators A: ll(iW-+
1£(IK)q and B: U -+ l£(lK)q by
(4.47) and B= (a,U)B.
, 1](https://image.slidesharecdn.com/1124525-250514194345-ce77afbf/85/Hilbert-Spaces-Of-Analytic-Functions-Crm-Proceedings-Lecture-Notes-Thomas-Ransford-49-320.jpg)
![34 J. A. BALL AND V. BOLOTNIKOV
Making use of the two last equalities and of (4.28) we get
q P q
L Q.k(Z)T[Aflk(z) = L:L:<l.ik(z)[AJ1k,j(Z)
k=l j=lk=l
P q P
=L:L:<l.ik(z)0(1 - Q(Z)A)-lI, L:Ak,x.
j=l k=l ,=1
P q P
= O(I - Q(Z)A)-1 LL<l.7k(Z)I, L:Ak,x,
j=l k==l 0=1
== 0(1 - Q(z)A)-lQ(z)Ax
which together with (4.50) implies (4.17). Similarly we conclude from (4.15), (4.45 •
(4.47) and (4.43) that
q P q
LQ.k(Z)T[Bu]k(Z) = LL<l.ik(z)[Bku],(z)
k=l j=lk=l
P q
= L L <l.ik(z) [G(Z)BkU]j
j=lk=1
P q
= L L <l.ik(Z)O(I - Q(Z)A)-lI,Bku
j=lk=l
P q
= 0(1 - Q(Z)A)-l L L <l.7k(z)I,Bku
j=lk=l
= 0(1 - Q(z)A)-lQ(z)Bu = S(z)u - S(O)u
and thus, B solves the Gleason problem (4.18) for S. Finally, for f and:c of the
form (4.49), for the operator U defined in (4.46) and for the operator C defined on
1l(lK)P by formula (4.29), we have
P P P P
C(EBU)x = Cf =Lik,k(O) = L O(I - Q(O)A)-lIkXk =LOIkXk =ex,
i=l k=l k=l k=l
and thus,
C(~U) =0.
The latter equality together with (4.47) implies (4.39). According to Definition 4.7,
the colligation U = [~ g] is a canonical functional-model colligation associated
with the Agler decomposition lK of S. 0
Let us say that the operator A: 1l(lK)P --+ 1l(IK)q is a contractive solution of the
Q-coupled Gleason problem for 1l(1K) if in addition to identity (4.17) the inequality
k
IAfll~(K)4 ~ IIfl~(K)P - Llfk,k(O)II~
, 1
holds for every function f e 1l(lK)P or equivalently, if the pair (C, A) is contractive
where C: 1l(lK)P ~ Y is the operator given in (4.29).](https://image.slidesharecdn.com/1124525-250514194345-ce77afbf/85/Hilbert-Spaces-Of-Analytic-Functions-Crm-Proceedings-Lecture-Notes-Thomas-Ransford-51-320.jpg)
![TRANSFER-FUNCTION REALIZATION 36
Theorem 4.10. Let (4.4) be tl fixed Agler decomposition of tl given function
S E SAQ(U, Y) and let 0 and D be defined as in (4.29). Then
(1) For every contractive solution A of the Q-coupled Gleason problem (4.17),
there is an operator B: U ~ 1i(lK)q such that U = l~ ~1is a canonical
functional-model colligation for S.
(2) Every such B solves the Gleason problem (4.18) for S.
The proof is very much similar to the proof of Theorem 3.10 and will be omitted.
In conclusion, we compare functional model Q-realizations obtained in this
section with particular cases considered in Sections 2 and 3.
The unit ball setting. In this case, Q = Zrow (in particular, p = 1) and def-
inition 2.2) can be interpreted as the (uniquely determined) Agler decomposition
of the form (4.4) with the kernellK= Ks. Then (4.23) gives 'll'(z,() = Ks(z,() and
4.33 coincides with (2.15). Since all canonical functional-model colligations are
obt.ained via contractive extensions of isometries V (from (2.15) for the unit ball
setting or from (4.33) for the general Q-setting), it follows that realizations con-
structed in Section 2 ca,D be obtained from those in Section 4 by letting Q = ZIOW'
Moreover, if Q = Zrow, then observability in the sense of (4.38) collapses to ob-
servability defined in part (2) of Theorem 2.9.
The unit polydisk setting. In this case, Q = Zdiag, P = q = d, and the
Agler representation (3.1) for an S E SAd(U,y) can be written in the form (4.4)
Kl 0
with the kernel ][{ = • • •
4.51
. Then (4.23) takes the form
'll'(Z, () = •
•
•
Kd(Z, () I8l ed
where {eh--' ,ed} is the standard basis for ICd. Observe that (4.51) is not the same
as 3.10 . Now 4.33) collapses to
(lKl(Z, () I8l el Kl(Z, () I8l el
4.52) V:
•
•
•
(dKd(Z, () I8l ed
y
whereas 3.24) can be written as
(lK1(z,()
•
-
4.53 V:
,
•
(dKd(Z, ()
y
,
•
•
Kd(Z, () I8l ed
S«()"y
K1(z, ()
•
,
,
Kd(Z, ()
S«()*y
-
To get canonical functional-model realizations as in Definition 3.5, we extend V to
. - A" C" d ( d
a contractIOn U" = B" D" : EB.=l1i Ki ) EB Y ~ EBi=l1i(Ki) EB u. If for such a
contraction we let U to be of the form (3.2) with
- -
A., = A., I8l e.e;, OJ = OJ I8l ej,](https://image.slidesharecdn.com/1124525-250514194345-ce77afbf/85/Hilbert-Spaces-Of-Analytic-Functions-Crm-Proceedings-Lecture-Notes-Thomas-Ransford-52-320.jpg)
k,l
{Bu1 a = : where in turn (Bul~a) = : E ll(K(a»). We de-
• •
[Bu] " [Bu](a)
0: k,po
fine a canonical functional-model colligation matrix U for a given function S E
SAq U.Y and left Q-Agler decomposition {K1, ... ,Kd} (so (4.54) holds) to be
any perator-matrix U = {~gl: E9!=lll(K(a»)Pa ffiU -+ E9!=lll(K(a»)qa ffiY so
that
1 U is a contraction.
2 the operator A solves the Q-structured Gleason problem for the kernel
collection {K(I)•...• K(d)}.
3 the operator B solves the Q-structured {K(1), ... ,K(d)}-Gleason problem
for S, and
4 the operators C and D are given by
11
(a)
d d Pa
C:ffi : ~LLlt2(0), D:u~S(O)u.
a=1
'
(a)
Pc>
a=lk=1
Then we leave it to the industrious reader to check that Theorems 4.8, 4.9 and 4.10
all go through with this block-diagonal modification. Specializing this formalism
to the polydisk c.ase picks up exactly the results of Section 3.
References
1. J Agler, On the representatIOn of certain holomorphc functiona defined on a polydisc, Topics
in Operator ThcOIY: Ernst D. Hellinger Memorial Volume, Oper. Theory Adv. Appl., vol. 48,
, Barel, 1990, pp. 47 66.
2. J. Agler and J. E. McCarthy, Nevanl,nna PIck Interpolaton on the bid,sk, J. Reine Angew.
Math. 606 (1999), 191 204.](https://image.slidesharecdn.com/1124525-250514194345-ce77afbf/85/Hilbert-Spaces-Of-Analytic-Functions-Crm-Proceedings-Lecture-Notes-Thomas-Ransford-54-320.jpg)
![realism, bathing towels[6] were hung from the verandahs. People passing by,
and ignorant of the double meaning of the curious drapery, shrugged their
shoulders, scoffed—then, awakened by a flash of illumination, looked again
and broke into renewed cheers. Before the dwelling of the mother of the
defender of Mafeking a vast crowd collected, wielding flags and laurels, and
displaying in their midst the bust of the hero with a British lion crouching at
his feet. Cheers rent the air, and increased in volume when the proud parent
of this splendid Briton appeared on the balcony and acknowledged the
demonstration. The glad tumult in front of this point of attraction continued
throughout the day, people coming from far and wide here to vent their
ecstasy of enthusiasm—some in shouts, many in tears.
By nightfall, the whole Empire was pouring forth its excitement in
congratulatory telegrams, for, four minutes after the receipt of the intelligence
in London the news had passed over the Atlantic cables and was in the New
York office of the Associated Press, whence it was forwarded to the farthest
limit of the North American Continent. Canada, New South Wales, Sydney, and
all the other colonies whose bravest and best had contributed to the great
doings in the Transvaal, were now aglow with bunting and illuminations.
Church bells pealed, processions passed shouting and rejoicing, ships were
dressed from truck to taffrail, and prayers and anthems of praise were got
ready to be offered up on the following day at all churches.
Thus, for a brief space, was seen a vast concourse of millions of souls of
differing opinions, customs, and creeds, diffused even to the remotest corners
of the British-speaking world, yet closely united by a bond of fraternal
sympathy in consequence of the triumph of British manhood in the most
unique ordeal that the loyalty of any nation has been called upon to endure.
FOOTNOTES:
[5] See Vol. III. p. 39.
[6] The hero of Mafeking at Charterhouse was nicknamed “Bathing Towel.”](https://image.slidesharecdn.com/1124525-250514194345-ce77afbf/85/Hilbert-Spaces-Of-Analytic-Functions-Crm-Proceedings-Lecture-Notes-Thomas-Ransford-61-320.jpg)
![breaking open shops and stores and houses, and throwing out of doors and windows goods
collected for the benefit of needy burgher families. The uproar, however, was speedily
suppressed by the firm measures of Dr. Krause. In answer to the flag of truce sent in by the
Field-Marshal, this official went out to meet him. There being still many armed burghers in the
place, the Transvaal Commandant requested Lord Roberts to postpone his entry for six hours.
To avert disturbance this arrangement was agreed to, and Lord Roberts decided to postpone
till the 31st his entry into the conquered town.
So Johannesburg was ours! The advance, which appeared to be so rapid, straightforward, and
simple, owed these qualities to Lord Roberts’s splendid, almost prophetic, instinct for gauging
the enemy’s expectations with a view to disappointing them; to his strategic manipulation of
his cavalry and mounted infantry, and to the magnificent marching capability of the infantry.
Everywhere, the Boers had fenced themselves across the route, sometimes extending their
line of defence for twenty miles or more, and everywhere, in dread of having one flank or the
other turned, they had been kept oscillating between stubborn resistance and rapid flight till
their nerves had given way, and they had scuttled back and back to their undoing. At the Vet,
the Zand, the Valsch, the Rhenoster, and the Klip Rivers, they had cunningly prepared
themselves, till, with the infantry menacing them in front and the cavalry and infantry
threatening both flanks, they had realised that retreat was inevitable. Their last hope had
been set on the city of mines; and now from thence, a routed, raging rabble, they were
fleeing in despair.
The splendid progress of the infantry was a remarkable achievement, of which enough cannot
be said. It was no mere feat of pedestrianism. It was a march in face of an enterprising
enemy, and harassed with discomforts sufficiently multifarious to try the endurance of a
Socrates. A scorching sun by day and a frigid temperature by night, occasional sand blasts
rendering drier than ever parched throats already dry as husk from the tramp through a sand-
clogged and almost waterless country, were but items in the programme. If water there
chanced to be, it was ochreous and fouled by the passage of many quadrupeds, and such
food as there was—bully beef and adamantine biscuit—demanded the jaws and digestion of
an alligator. Yet these sturdy fellows plodded along, lumbering through sand drifts and
squelching in mire and morass, or laid themselves to rest on the hard or soggy ground with a
philosophy so devil-may-care as almost to fringe on the sublime. With unquenchable gaiety,
they had accomplished a march of 254 miles (the distance from Bloemfontein to
Elandsfontein) in eighteen days, giving as an average fourteen miles a day. (This calculation
naturally excludes the ten days’ halt at Kroonstad.) From Kroonstad to Elandsfontein, a
distance of some 126 miles—covered in seven days (22nd to 29th)—marching had gone
forward at the rate of eighteen miles a day. Napoleon’s much vaunted march from the
Channel to the Rhine in 1805 showed an average of sixteen miles a day, when the distance
traversed was 400 miles, and the time taken twenty-five days. But that march, unopposed
throughout, was comparatively plain sailing. Quicker forced marches have been known,[7] but
in the present case the march was continuous, and may be said to beat all records of rapid
marching under equally inconvenient conditions.
The twenty-four hours were allowed to pass. Then, at the entrance of the town Dr. Krause
met the Commander-in-Chief, and rode with him to the government offices, and introduced to
him the heads of the various departments, all of whom were requested to continue their
respective duties till they should be relieved of them.](https://image.slidesharecdn.com/1124525-250514194345-ce77afbf/85/Hilbert-Spaces-Of-Analytic-Functions-Crm-Proceedings-Lecture-Notes-Thomas-Ransford-70-320.jpg)
![Vaal River Deviation Bridge at Vereeniging, nearly completed.
(Photo by W. H. Gill, London.)
It was an impressive spectacle; one ever to be remembered. From afternoon till night, troops
—great, brawny, bronzed, and workmanlike Britons—came clanking in procession through the
town, while from balconies and windows banners and flags were waved, and gay ladies, many
of them Englishwomen, wild with excitement and enthusiasm, threw down flowers and sweets
and cigarettes to give vent to their unrestrained joy. Far into the evening the stream of kharki
continued ceaselessly to flow under the magnesian rays of the electric lights till the infantry
had passed to their camp, three miles to the north, and Lord Roberts had settled himself at
Orange Grove.
FOOTNOTES:
[7] See vol. iv. p. 41.](https://image.slidesharecdn.com/1124525-250514194345-ce77afbf/85/Hilbert-Spaces-Of-Analytic-Functions-Crm-Proceedings-Lecture-Notes-Thomas-Ransford-72-320.jpg)
Hilbert Spaces Of Analytic Functions Crm Proceedings Lecture Notes Thomas Ransford
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- 6. • -1 e aces 0 • une Ions
- 7. Preface The workshop entitled Hilbert Spaces of Analytic Functions was held at the Centre de recherches matMmatiques (CRM), Montreal, from 8 to 12 December 2008. Even though this event was not a part of the CRM thematic year, 62 math- ematicians attended the workshop. They formed a blend of researchers with a common inteIest in spaces of analytic functions, but seen from many different an- gles. lhlbert spaces of analytic functions are currently a very active field of com- plex analyslS. The Hardy space H2 is the most senior member of this family. Its relatlVes, such as the Bergman space AP, the Dirichlet space V, the de Branges- Rovnyak spaces 1£ b), and various spaces of entire functions, have been extensively studIed by prominent mathematicians since the beginning of the last century. These spaces ha been explOlted in different fields of mathematics and also in physics and engmeering. For example, de Branges used them to solve the Bieberbach conjec- ture, and Zames, a late professor of McGill University, applied them to construct hlS th ry f HOC control. But there are still many open problems, old and new, which attract a wid spectrum of mathematicians. In thlS nference, 38 speakers talked about Hilbert spaces of analytic functions. In five days a wi e vanety of applications were discussed. It was a lively atmosphere m which many mutual research projects were designed. xi Javad Masbreghi Thomas Ransford Kristian Seip
- 10. The production of this volume was supported in part by the Fonds Quebecois de la Recherche sur la. Nature et les Technologies (FQRNT) and the Natural Sciences aDd Engineering Research Council of Ca.nada (NSERC). 2000 Mathematics Subject Classification. Primary 46E20, 46E22, 47B32, 31C25. Library of Congress Cataloging-In-Publication Data Hilbert spaces of analytic functions / Javad Masbreghi, Thomas Ransford, Kristian SeIP, echtora. p. cm. - (CRM proceedings & lecture notes; v. 51) Includes bibliographical references. ISBN 978-0-8218-4879-1 (alk. paper) 1. Hilbert space-Congresses. 2. Analytic functions-Congresses. 3. P tentJal theory (Mathematics)-Congresses. I. Mashreghi, Javad. 11. Ransford, Thomas. 111. 8eJ.p, Knsban., 1962- QA322.4.H55 2010 515'.733---dc22 2010003337 Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduCtIon by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copymg for general distribution, for advertising or promotional purposes, or for resale. Requests for permisl!lon for commercial use of material should be addressed to the Acquisitions Department, American Math- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made bye-mail to reprint-perlllissionGams .org. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author s (Copyright ownership is indicated in the notice in the lower right-hand comer of the first page of each article.) © 2010 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Copyright of individual articles may revert to the public domain 28 years after publication. Contact the AMS for copyright status of individual articles. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. This volume was submitted to the American Mathematical Society In camera ready form by the Centre de Recherches Mathem.atiques. Visit the AMS home page at http://lnIV. ama.org/ 10 9 8 7 6 5 4 S 2 1 15 14 13 12 11 10
- 11. Volume 51 Centre de Recherches Mathematlques Montreal • 1 Javad Mashreghi Thomas Ransford Kristian Seip Editors aces 0 The Centre de Recherches Mathematiques (CRM) of the Universire de Montreal was created in 1968 to promote research in pure and applied mathematics and related disciplines. Among its activities are special theme years, summer schools, workshops, postdoctoral programs, and publishing. The CRM is supported by the Universite de Montreal, the Province of Quebec (FQRNT), and the Natural Sciences and Engineering Research Council of Canada. It is affiliated with the lnstitut des Sciences Mathematiques (ISM) of Montreal. The CRM may be reached on the Web at www.clm.math.ca. American Mathematical Society Providence. Rhode Island USA
- 12. List of Participants List of Speakers Contents Canonical de Branges Rovnyak Model 'Iransfer-Function Realization for Multivariable Schur-Class Functions .. Vll • IX • Xl Joseph A. Ball and Vladimir Bolotnikov 1 Two Variations on the Drury Arveson Space NUXJla Arcozzi, Richard Rochberg, and Eric Swayer 41 The Norm of a Truncated Toeplitz Operator Stephan Ramon Garcia and William T. Ross 59 Approximation in Weighted Hardy Spaces for the Unit Disc Andre Bo vin and Changzhong Zhu 65 Some Remarks on the Toeplitz Corona Problem Ronald Douglas and Jaydeb Sarkar 81 on the BOllndary in Spaces of Holomorphic Functions on the Unit Disc Emmanuel Fricain and Andreas Hartmann 91 The Search for Singula.rities of Solutions to the Dirichlet Problem: Recent Developments Dmitry Khavinson and Eric Lundberg 121 Invariant Subspaces of the Dirichlet Space Omar El-Fallah, Karim Kellay, and Thomas Ransford 133 Arguments of Zero Sets in the Dirichlet Space Javad Mashreghi, Thomas Ransford, and Mahmood Shabankhah 143 Questions on Volterra Operators Jaroslov Zemanek 149 Nonhomogeneous Div-Curl Decompositions for Local Hardy Spaces on a Domain Der-Chen Chang, Galia Dafni, and Hong Yue 153 On the Bohr Radius for Simply Connected Plane Domains Richard Fournier and Stephan Ruscheweyh 16E
- 13. vi CONTENTS Completeness of the System {/(AnZ)} in L�[nJ Andre Boivin and Changzhong Zhu A Formula for the Logarithmic Derivative and Its Applications Javad Mashreghi Composition Operators on the Minimal Mobius Invariant Space Hasi Wulan and Chengji Xiong Whether Regularity is Local for the Generalized Dirichlet Problem Paul M. Gauthier 173 197 203 211
- 14. List of Participants Evgueni AbakOllmov Universite Paris-Est Fatma Z. Abdessameud Universite Saad Dahlab Blida Jim Agler UC San Diego Thyeb Ai"ssiou McGill University Nadya Askanpour University of Western Ontario Hirbod Assa. Universite de Montreal Amoah G. AtingabllDor Khomananj College Joseph A. Ball VIrginia Tech Ferenc Balogh Concordia University Laurent Baratcbart INRlA Sophia Antipolis-Mediterranee Andre Boivin University of Western Ontario Alexander Borichev Universite de Provence Abdellatif Bourbim Universite Laval Marcus Carlsson Purdue University Nicolas Chevrot Universite Laval Yemon Choi Universite Laval Joseph A. Cima UNC Chapel Hill Constantin Costara Universitatea Ovidius Galla Dafni Concordia University Ronald G. Douglas Texas A&M University Omar EI-Fallah Universite Mohammed V Yasser Farhat Universite Laval Tatyana Foth University of Western Ontario Richard Fournier Universite de Montreal Emmanuel Fricain Universite Claude Bernard Lyon 1 Frederic Gaunard Universite Bordeaux 1 Paul M. Gauthier Universite de Montreal Kohur Gowrisankaran McGill University Dominique Guillot Universite Laval Andreas Hartmann Universite Bordeaux 1 vii
- 15. vIII Tarik Jari Universite Laval Dmitry Khavinson University of South Florida Daniela Kraus Universitii.t Wiirzburg Alex Louis University of Swaziland Yura Lyubarskii NTNU Mostafa Mache Universite Laval Davood Malekzadeh Universite Laval Jordi Marzo NTNU Javad Mashreghi Universite Laval Tesfa Mengestie NTNU Putinar Mihai UC Santa Barbara Joules Nahas UC Santa Barbara Mostafa Nasri Universite Laval Joaquim Ortega-Cerda. Universitat de Barcelona Vladimir Peller Michigan State University Mihai Putinar UC Santa Barbara Quentin Rajon Universite Laval Thomas J. Ransford Universite Laval Richard Rochberg PARTICIPANTS William T. Ross University of llichmond Stephan Ruscheweyh Universitat Wiirzburg Sergey Sadov Memorial University of'iewfoundland Kristian Seip NTNU Alexis Selezneff Universite Laval Mahmood Shabankhah Universite Laval Richard Spjut UC Santa Barbara Qingyun Wang Washington Univers ty in St. Lows Hasi Wulan Shantou Um erslty Nicolas Y ung Leeds University HongYue ConcordIa University Jaroslav Zemanek Polish Academy of SCIences Nina Zorboska University of Manitoba Washington University in St. Louis
- 16. List of Speakers Evgueni Abakllmov Universite Paris-Est Fatma Z. Abdessameud Universite Saad Da.hlab Blida Jim Agler UC San Diego Joseph A. Ball Virginia Tech Ferenc Balogh Concordia University Laurent Baratchart INRIA Sophia Antipolis - Meditarranee Andre Boivin University of Western Ontario Alexander Borichev Universite de Provence Abdelatif Bourbim Universite Laval Nicolas Chevrot Universite Laval Joseph Cima UNC Chapel Hi11 Constantin Costara Universitatea Ovidius Ga.1ia Dafni Concordia University Ronald G. Douglas Texas A&M University Omar EI-Fallah Universit.e Mohammed V Tatyana Foth University of Western Ontario Richard Fournier Universite de Montreal Emmanuel Fricain Universite Lyon 1 Paul Gauthier Universite de Montreal Kohur GowriSanka.ran McGill University Andreas Hartmann Universite Bordeaux 1 Dmitry Khavinson University of South Florida Danieal Kraus Universitat Wiirzburg Yura Lyubarskii NTNU Jordi Marzo NTNU ix Joules Nahas UC Santa Barbara Joaquim Ortega-Cerda Universitat de Barcelona Vladimir Peller Michigan State University Mihai Putinar UC Santa Barbara Thomas Ransford Universite Laval
- 17. x Richard Rochberg Washington University in St. Lo William T. Ross University of Richmond Stephan Ruscheweyh Universitiit Wiirzburg Mahmood Shabankhah Universite Laval Hassi Wulan Shantou University Nicolas Young University of Leeds Hong Yue Concordia University Jaroslav Zemanek Polish Academy of Science
- 18. C~nt.. de Recherchee Math~matlqu"" CRM Proceedingl and J ecture Notea Volume &1. 2010 Canonical de Branges - Rovnyak Model Transfer-Function Realization for Multivariable Schur-Class Functions Joseph A. Ball and Vladimir Bolotnikov ABSTRACT. Associated with any Schur-c111S8 function S(z) (i.e., a contractive holomorphic function on the the unit disk) is the de Branges-Rovnyak kernel Ks{~, C) = [1-S{z)S(C)·]/(l- z() and the de Branges- Rovnyak reproducing kernel Hilbert spsce 1i(Ks). This space plays a prominent role in system theory as a canonical-model state space for a transfer-function realization of a given Schur-class function. There has been recent work extending tbe notion of Schur-cJ1IRS function to several multivariable settings. We bere ma.ke explicit to what extent the role of de Branges-Rovnyak spaces as the canonical-model state space for transfer-function realizations of Schur-class functions extends to multivariable settings. 1. Introduction Let U and Y be two Hilbert spaces and let L(U, Y) be the space of all bounded linear operators between U a.nd y. The operator-valued version of the classical Schur class S(U,Y) is defined to be the set of all holomorphic, contractive L(U, Y)- valued functions on 1Il. The following equivalent characterizations of the Schur class are well known. Here we use the notation H2 for the Hardy space over the unit disk and ~ = H2 ® X for the Hardy space with values in the auxiliary Hilbert space X. Theorem 1.1. Let S: 1Il -+ L(U, Y) be given. Then the following are equiva- lent: (1) (a) S E S(U, Y), i.e., 8 is holomorphic on 1Il with 118(z)11 ~ 1 for all z E 1Il. (b) The opemtor Ms: fez) I-t 8(z)f(z) of multiplication by 8 defines a contmction opemtor from H~ to H~. (c) S satisfies the von Neumann inequality: 118(T)1I ~ 1 for any strictly contmctive opemtor T on a Hilbert space 11., where 8(T) is defined 2000 Mathellwzbcs Subject ClasBifico.tion. 47A57. Key words and phrasea. Operator-valued functions, Schur multiplier, canonical functional model. reploducing kernel Hilbert space. This is the final {olm of the paper. @20l0 American Mathematical Society 1
- 19. 2 (1.1) J. A. BALL AND V. BOLOTNIKOV by 00 'Xl SeT) = L Sn ® rn E C(U ® 1/., Y ® 1/.) if S(z) = LSnz'l. n=O (2) The associated kernel Ks(z, () = Iy - S(z)~«t 1-z( n=O is positive on JD) x JD), i.e., there exists an operator-valued functwn H. D-+ C(X, Y) for some auxiliary Hilbert space X so that (1.2) Ks(z, () = H(z)H«t· (3) There is an auxiliary Hilbert space X and a umtary conned ng operator so that S(z) can be expressed as (1.3) S(z) = D +zO(I - zA)-lB. (4) S(z) has a realization as in (1.3) where the connect ng opemtorU u one of (i) isometric, (ii) coisometnc, or (iii controch e. We note that the equivalence of any of (la), (lb, 1c with 2 and can be gleaned, e.g., from Lemma V.3.2, Proposition 1.8.3, Proposition V.S.1 and The- orem V.3.1 in [26]. As for condition (4), it is trivial to see that 3 imphe:s 1 and then it is easy to verify directly that (4) implies (la . Alternatl,-ely, one can use Lemma 5.1 from Ando's notes [6] to see directly that 4iii implies .fu see Remark 2.2 below). The reproducing kernel Hilbert space 1/.(Ks) with the de Branges-Rorn ilk kernel Ks(z, () is the classical de Branges - Rovnyak reproducmg kernel Hrlberl space associated with the Schur-class function S which has been much studied O-ef the years, both as an object in itself and as a tool for other types of applications see [6,11-13,16-18,20,21,24,27,28,31]). The special role of the de Branges-Ro'Il'8k space in connection with the transfer-function realization for Schur-class functIons is illustrated in the following theorem; this form of the results appears at Iea:. 1 implicitly in the work of de Branges Rovnyak [20,21]. Theorem 1.2. Suppose that the function S is in the Schur class S(U, y and e 1-£(Ks) be the associated de Branges Rovnyak space. Define operators A,B,C,D by A: fez) 1-+ fez) - f(O) , z C: fez) 1-+ f(O), B: u I-? S(z) - S(O) u, z D: u I-? S(O)u. Then the operator-block matrix U = l~ ~] has the following properties: (1) U defines a coisometry from 1/.(Ks) e U to 1/.(Ks) e y. (2) (C,I) is an observable pair, i.e., CAn f ... 0 for all n =0,1,2, ... """'=> J =0 as an element of1-£(Ks). (3) We recO'Uer S(z) as S(z) = D ;.. zO(I _ .tA) lB.
- 20. TRANSFER-FUNCTION REALIZATION (4) If [~: g:1:X Ell U -+ X ffi Y is another colligation matrix with properhes (1), (2), (3) above (with X in place of 1£(Ks)), then there is a unitary operator U: 1£(Ks) -+ X so that U 0 A B A' B' U 0 - o Iy C D - C' D' 0 Iu' It is easily seen from characterization (la) in Theorem 1.1 that (1.4) S E S(U,Y) < ;- S E S(Y,U) where S(z) := S(i)*. Hence for a given Schur-class function S there is also associated a dual de Branges- Rovnyak space ?-leKs) with reproducing kernel Ks(z, () = [1-8(i)*S(())/(l-z(). The space ?-leKs) plays the same role for isometric realizations of 8 as ?-l(Ks) plays for coisometric realizations, as illustrated in the next theorem; this theorem is just the dual version of Theorem 1.2 upon application of the transformation (1.4). Theorem 1.3. Suppose that the function S is in the Schur class S(U,Y) and let ?-leKs) be the associated dual de Branges-Rovnyak space. Define operators Ai, Bd, Cd, Dd by A.i: g(z) 1-+ zg(z) - S(i)* yeO), Bd: u 1-+ (I - S(i)*S(O))u, Cd: g(z) 1-+ yeO), Dd: u 1-+ S(O)u, whel e y 0) I.S the unique vector in Y such that -(O)} ( ) S(i)* - S(O)* 9 ,y y = 9 z , Y z for all y E y. Then the operator-block matrix U d = l~: g~1has the following properties: 1) Ud defines an isometry from ?-leKs) Ell U to 'tl.(Ks) Ell y, 2 (Ai,Bd) is a controllable pair, i.e., Vn>oRanA'dBd = 'tl.(Ks), where V - stands for the closed linear span. 3 We recover S(z) as S(z) = Dd +zCd(I - ZAd)-lBd' 4) If [~' g',1 :X Ell U -+ X Ell Y is another colligation matrix with properties (1 , (2), (3) above (with X in place of 1£(Ks )), then there is a unitary operator U: 'tl.(Ks) -+ X so that U 0 Ad Bd A' B' U 0 - o Iy Cd Dd - C' D' 0 Iu' ~ In addition to the kernels Ks and Ks , there is a positive kernel Ks which combines two and is defined as follows: (1.5) ~ Ks(z,() K(z, () = S(z)-S(q %-c ~ S(%)-S(~) %-, Ks(z, () I-S(z)8JQ·, _ 1-%~ - S z o_s C• %- I -s(.;:) 8m . 1-ol( ol- It tW"IIS out that K is also a positive kernel on Jl)) x Jl)) and the associated reproducing ~ kernel Hilbert space ?-l(Ks) is the canonical functional-model state space for unitary realizations of 8 , as summarized in the following theorem. This result also appears at least implicitly in the work of de Branges and Rovnyak [20,21J and more explicitly ~ the paper of de Branges and Sbulman [22J, where the two-component space ?-leKs) associated with the Schur-clfl13l3 function S is denoted as D(S); see also [11J for an explanation of the connections with the Sz.-Nagy-Foias model space.
- 21. 4 J. A. BALL AND V. BOLOTNIKOV Theorem 1.4. Suppose that the function S is in the Schur class S U,y) aM let K(z,() be the positive kernel on JI} given by (1.5). Define operators A,B,c fi by A' rf(z)1 t-+ r[fez) - f(0)1Iz 1 . Lg(z) Lzg(z) - S(z)f(O) , B ~ . f([S(z) - S(O 1z u1 . u I-t l (I - S(z)*SOu 8: rf(z)1 t-+ f(O) Lg(z) , D: u I-t S(O)u. Then the operator-block matrix U = [~ ~1satisfies the following: (1) Udefines a unitary operator /rom 1£(Ks) G3 U onto 1£ Ks G3 Y· (2) U is a closely connected operator colligation, i.e., V{Ran..4.nB, Ran..4.*nC*} = 1£(Ks). n~O (3) We recover S(z) as S(z) = D+zC(I - z..4.)-1B. (4) If [~: g:] :X G3 U -+ X G3 Y is any other opemtor colligatwn sahsfym9 conditions (1), (2), (3) above (with X in place of1£ Ks , then there IS unitary operator U: 1£(Ks) -+ X so that rU 01 fA Bl fA' B'1 ru 01 L 0 Iy lC DJ= L 0' D' lo Iu' Our goal in this article is to present multivariable analogues of Theorem 1.2. The multivariable settings which we shall discuss are (1) the unit ball Bel in C' and the associated Schur class of contractive multipliers between vector-valued Drury Arveson spaces lI.u(kd) and 11.y (kd), (2) the polydisk with the associated Schurclass taken to be the class of contractive operator-valued functions on Dei which satlsfy a von Neumann inequality, and (3) a more general setting where the underlying domain is characterized via a polynomial-matrix defining function and the Schur class is defined by the appropriate analogue of the von Neumann inequality. In the'le multivariable settings, the analogues of Theorem 1.1 have already been set down at length elsewhere (see [3,15,23] for the ball case, [1,2,14] for the polydisk case, and [4,5,9] for the case of domains with polynomial-matrix defining function-see [8] for a survey). Our emphasis here is to make explicit how Theorem 1.2 can be extended to these multivariable settings. While the reproducing kernel spaces themselves appear in a straightforward fashion, the canonical model operators on these spaces are more muddled: in the coisometric case, while the analogues of the output operator 0 and the feedthrough operator D are tied down, there is no canonical choice of the analogue of the state operator A and the input operator B: A and B are required to solve certain types of Gleason problems; we refer to [25] and [30, Section 6.6] for some perspective on the Gleason problems in general. The Gleason property can be formulated also in terms of the adjoint operators A' and B': the actions of the adjoint operators are prescribed on a certain canonically prescribed proper subspace of the whole state space. From this latter formulation, one can see that the Gleason problem, although a.t first sight a.ppearing to be rather complicated, always has solutions. Also, the adjoint of the colligation matrix, rather than being isometric, is required only to be isometric on a certain subspace of the whole space X EB y. With these adjustments, Theorem 1.2 goes through in the
- 22. TRANSFER-FUNCTION REALIZATION 5 three settings. Most of these results a.ppear in [10] for the ba.ll case and in more implicit form in [14] for the polydisk case, although not in the precise formulation presented here. The parallel results for the third setting are presented here for the first time. We plan to discuss multivariable analogs of Theorems 1.3 and 1.4 in a. future publication. The paper is organized as follows. After the present Introduction, Section 2 lays out the results for the ball case, Section 3 for the polydisk case, and Section 4 for the case of domains with polynomial-matrix defining function. At the end of Section 4 we indicate how the results of Sections 2 and 3 can be recovered as special of the general formalism in Section 4. 2. de Branges-Rovnyak kernel associated with a Schur multiplier on the Drury - Arveson space A natural extension of the Szego kernel is the Drury - Arveson kernel 1 1 kd(Z,() = =: ( . 1-Z1(1-"'-Zd(d 1- Z,()Cd The kernel kd Z, () is positive on Bd x Bd where Bd = {z = (Zl' ... , Zd) E Cd: (z, z) = Zl2 +... + Zd2 < I} is the unit ball in Cd, and the associated reproducing kernel Hilbert space ll(kd) is called the Drury-Arveson space. For X any auxiliary Hilbert space, we use the shorthand notation llX(kd) for the space ll(kd) ® X of vector-valued Drury- Arveson-space functions. A holomorphic operator-valued function S: Bd -+ LeU, y) is to be a Drury Arveson space multiplier if the multiplication operator Ms: / z t--+ S Z fez) defines a bounded operator from llU(kd) to lly(kd)' In case in addition Ms defines a contraction operator (IMslop 5 1), we say that S is in the Schur-multiplier class Sd(U, Y). Then the following theorem is the ana.- logue of Theorem 1.1 for this setting; this result appears in [10,15,23). The alert reader will notice that there is no analogue of condition (la) in Theorem 1.1 in the following theorem. Theorem 2.1. Let S: Bd -+ l.(U, Y) be given. Then the following are equiva- lent: 2.1) 1) (b S E Sd(U, Y), i.e., the operator Ms of multiplication by 8 defines a contraction operator from llU(kd) into lly(kd)' c) S satisfies the von Neumann inequality: 8(T)1 5 1 for any com- mutative operator d-tuple T = (Tl' ... ,Td) of operators on a Hilbert space /C such that the operator-block row matrix [Tl . .. Td] defines a stnct contraction operator from /Cd into /C, where SeT) = L Sn®-rn E l.(U®ll,Y®ll) nEZi if 8(z) = L 8n zn . nEZd + Here we use the standard multivariable notation: z'" = Z~1 .,. Z~d and ~ = 'If1 ...T:;d if n = (nl," ., nd) E Zi. (2) The associated kemel (2.2) K ( 1") = 1y - 8(z)8«>* s z,... 1 _ (z, ()
- 23. 6 J. A. BALL AND V. BOLOTNIKOV is positive on B xB, i.e., there exists an opemtor-valued function H. B.t -+ C(X, Y) for some a'I.IXiliary Hilbert space X so that Ks(z, <) = H(z H <• (3) There is an a'I.IXiliary Hilbert space X and a unitary connectmg operator (2.3) u- [~ ~l= l~ ~1' [~1-+l~ so that 8(z) can be expressed as (2.4) 8(z) =D +0(1 - Zrow(Z)A)-l Zrow(z)B, where we have set Zrow(z) = (zlIx ... zd1x). (4) 8(z) has a realization as in (2.4) where the connecting opemtorU as an one of (i) isometric, (ii) coisometric, or (iii) contmctwe. Remark 2.2. Statement (4iii) concerning contractive realizations is not men- tioned in [15] but is discussed in [10,23]. The approach in [231 is to show that for 8 of the form (2.4) with U = [~ g] contractive the inequality S T) S 1 h ds for any commutative operator d-tuple T = (Tlo ••• , Td with (Tl ..• Td1 <I, Le, one verifies (4iii) :::::} (Ie). The idea of the second approach in [10] is to embed the contraction U ={~g into a coisometry [~ ~] = (~ gg~] with associated transfer function of the form. S(z) = (8(z) 81(z)] equal to an extension of S(z) with a larger input space. From the coisometry property of [~~] one sees that Ks(z,w) = 0 1 -Zrow z A -1 1- A·Zrow«)-IC·, i.e., S meets condition (2) for the Schur-class Sd U eUhY With H(z) =C(l - Zrow(z)A)-l. From the equivalence (lb) ~ (2 , it is easy no" to read off that 8 E Sd(U, y). A third approach worked out for the classical case but extendable to multr variable settings appears in Ando's notes [6, Lemma 5.11. Given a contractiw colligation U = [~g], one can keep the input and output spaces the same but enlarge the state space to construct a coisometric colligation U = [~~1having the same transfer function, namely: r Qll Q12 0 0 ..] n=m' • 0 0 0 I 0 ... A= 0 0 0 0 I , c= (0 Q21 Q22 0 o ...], D=D where Q = [~i~ ~~~] =(I - UU*)1/2. In this way one gets a direct proof of (4iii) ==> (4ii). For colligations U of the form (2.3), it turns out that a somewhat weaker notior. of coisometry is more useful than simply requiring that U be coisometric.
- 24. TRANSFER·FUNCTION REALIZATION 7 Definition 2.3. The operator-block matrix U of the form (2.3) is weakly coiso- metric if the restriction of U" to the subspace (2.5) is isometric. Du-:= V (EBcI lEY Zrow«)*(l - A* Zrow«)*)-lO"y C Xd Y Y It turllS out that the weak-coisometry property of the colligation (2.3) is exactly what is needed to guarantee the decomposition (2.6) Ks(z,() = O(l - Zrow(z)A)(l - A"Zrow«)*)-lC* of the de Branges-Rovnyak kernel Ks associated with S of the form (2.4) (see Proposition 1.5 in [10]). 2.1. Weakly coisometric canonical functional-model colligations. As the kernel Ks given by (2.2) is positive on B x B, we can associate a reproducing kernel Hilbert space 1l(Ks) just as in the classical case, where now the elements of 1l Ks are holomorphic Y-valued functions on Bd. In the classical case, as we from Theorem 1.2, there are canonically defined operators A, B, C, D so that the operator-block matrix U = [~~l is coisometric from ll(Ks) ffiU to ll(Ks) ffi Y and yields the essentially unique observable, coisometric realization for S E S U,y. For the present Drury-Arveson space setting, a similar result holds, but the operators A, B in the colligation matrix U are not completely uniquely determined. To expJajn the result, we say that the operator A: ll(Ks) -+ 1l(KS)d s I es the Glea-c;on problem for 1l(Ks) if the identity d 2.7 / z) - f(0) = L zk(Afh(z) holds for all f E 1l(Ks), k=l (Alh (,,) where we write Af) z) = • • • E 1l(KS)d. We say that the operator B: U-+ (AIM") 1l Ks d solves the 1l(Ks)-Gleason problem for S if the identity d 2.8 S z)u - S(O)u = LZk(Bu)k(Z) holds for all u E U. k=l Solutions of such Gleason problems are easily characterized in terms of adjoint operators. Proposition 2.4. The operator A: ll(Ks) -+ 1l(KS)d solves the Gleason prob- lem for ll(Ks) (2.7) if and only if A*: 1l(KS)d -+ ll(Ks) has the following action all. special kernel functions: 2.9) A*: Zrow«)*Ksh ()y ~ Ks(o, ()y - Ksh O)y for all (E Bd , Y E y. The operator B: U -+ 1l(KS)d solves the ll(Ks)-Gleason problem for S (2.8) if and only if B*: 1l(KS)d -+ U has the following action on special kernel functions: (2.10) B*: Zrow«)*Ks("()Y t-+ S«)*y - S(O)"y for all (E Bd , Y E y.
- 25. 8 J. A. BALL AND V. BOLOTNIKOV PROOF. By the reproducing kernel property, we have for I E l£(Ks), (I(Z) - f(O),y}y = (f, Ks("z)y - Ks(·,O)Y}1£(Ks)· On the other hand, (t,zk(Af)k(Z), y))I = t«Af)k' ZkKS(" Z)Y}1£(Ks =(AI, Zrow(Z)* Ks(-, Z)Y}1£ Ks d =(f,A*Zraw(z)*Ks("z)y 1£(Ks and since the two latter equalities hold for all I E 1l(Ks), z E Bd and Y E :v the equivalence of (2.7) and (2.9) follows. Equivalence of (2.8) and 2.10 fol1~ similarly from the computation: (t,zk(Bu)k(z), y))I = t,«Bu)k, ZkKS(" z)y 1£ Ks = (Bu, Zrow(z)* Ks(-, z Y 1£ Ks = (u, B* Zraw(z)* Ks("z)y u· Let us introduce the notation (2.11) v = V Zrow«)"Ks(" ()y. t;Eud yE)I o Definition 2.5. Given S E Sd(U, y), we shall say that the block-operoto! matrix U = [~ ~1is a canonical functional-model colligation for S if (1) U is contractive and the state space equals 1l(Ks}. (2) A: l£(Ks) -+l£(Ks)d solves the Gleason problem for 1l Ks (2.7. (3) B: U -+l£(Ks)d solves the 1l(Ks)-Gleason problem for S 2.8). (4) The operators C: 1l(Ks) -7- Y and D: U -7- Yare given by (2.12) C: I(z) 1-+ 1(0), D: U 1-+ S(O)u. Remark 2.6. It is useful to have the formulas for the adjoints C" : Y -+11. Ks and D: Y -+U: (2.13) C*: y 1-+ Ks(-, O)y D* : y 1-+ S(O)*y which are equivalent to (2.12). The formula for D· is obvious while the formula. for C· follows from equalities (f,C·yhi(Ks) =(C/,y}Y = (f(O),y}y = (f,Ks(·,O)Y}1£(Ks) holding for every f E l£(Ks) and y E y. Theorem 2.7. There exists a canonical functional-model realization for etlery 5 E Sd(U,y). PROOF. Let 5 be in Sd(U, Y) and let 1l(Ks) be the associated de Branges Rovnyak space. Equality (2.2) can be rearranged as d L z,(,Ks(z, () +1)/ = Ks(z, () + S(z)S«)* ;-1
- 26. TRANSFER·FUNCTION REALIZATION which in turn, can be written in the inner-product form 88 the identity (2.14) Zraw«()* Ks(', ()y y I Zrow(z)*Ks(" Z)y' y' 'H(Ks)ci$)! - K s (', ()y Ks(', Z)y' - 9 S«(ty , S(Zty' 'H(Ks)$U holding for every y, y' E Y and (, z E Bd. The latter identity tells us that the linear map (2.15) extends to the isometry from Vy = V EB Y C 1i(KS)d EB Y (where V is given in (2.11)) onto .,., - V Ks(', ()y (K) U "-y - S«()*y C 1i s EB • CESci liE)! Extend V to a contraction U .. : 1i(KS)d EB Y ~ 1i(Ks) EBU. Thus, 2.16) U· - A* C* . Zrow«()*Ks(" ()y ~ Ks(', ()y - B* D* . y S«()*y . Comparing the top and the bottom components in (2.16) gives 2.17 A*Zrow«()*K s (" ()y +C*y = Ks(" ()y, 2.18 B*Zrow«()*K s (" ()y +D*y = S«()*y. Solving (2.17 2.19) for Ksc-, ()y gives Ks("()Y = (1 - A*Zrow«()*)-lC"y. Substituting this into (2.18) then gives 2.20 B*Zrow«()*(1 - A* Zraw«(t)-lC*y +D*y = S«(ty. By taking adjoints and using the fact that ( E Ba and y E Yare arbitrary, we may then conclude that U is a contractive realization for S. It remains to show that U meets the requirements (2) - (4) in Definition 2.5. To this end, we let ( = 0 in 2.17) and (2.18) to get (2.21) C*y = Ks("O)y and D*y = S(O)*y. Substituting (2.21) back into (2.17) and (2.18), we get equalities (2.9) and (2.10) which are equivalent (by Proposition 2.4) to A and B solving the Gleason prob- lems (2.7) and (2.8), respectively. By Remark 2.6, equalities (2.21) are equivalent to (2.12). 0 Remark 2.8. A consequence of the isometry property of V in (2.15) is that fot IIIlIlas (2.9) and (2.10) extend by linearity and continuity to give rise to uniquely detelll.ined well-defined linear operators Ai> and BD from V to 1i(Ks) and U, respectively. In this way we that the existence problem for operators A solving the Gleason problem is settled: A: l£(Ks) ~ l£(Ks)d solves the Gleason problem forl£(Ks) (2.7) if and. only if A" is an extension to all of1i(Ks)d of the opemtor Ai,: V ~ 1i(Ks) uniquely detennined by the formula (2.9). Similarly, the opemtor B: U ~ 1i(KS)d is a solution of the 1i(Ks)-Gleason problem for S (2.8) if and only
- 27. 10 J. A. BALL AND V. BOLOTNIKOV if the operator B*: ll(KS)d -+ U is an extension to all of ll(KS)d of the 0Jl"TIIt9I' BD:V -+ U uniquely determined from the fonnula (2.10). The following result is essentially contained in {IO}. For the ball setting, we 1l&! the following definition of observability: given an operator pair C, A WIth 0 tp operator C: X -+ Y and with A: X -+ X d , we say that C, A is obsenable If C(I - Zrow(z)A)-lx = 0 for all z in a neighborhood of 0 in Cd implies that % = in X. Equivalently, this means that V(I - A*Zrow(zt)-lC·y = X "EL!. yEY for some neighborhood 6. of 0 in Cd. Theorem 2.9. Let S be a Schur-class multzpher in Sd U,)J and suppose tha: U =[~ ~ 1is any canonical functional-model colhgatlon for S. Then: (1) U is weakly coisometric. (2) The pair (C, A) is observable. (3) We recover S(z) as S(z) = D +C(l - Zrow(Z A -lZrow Z B (4) If U' = [~: g:1:X E9 U -+ Xd E9 Y is any other II gahon matn:r en- joying properties (1), (2), (3), then there is a canon cal functw'ool- colligation U = [~ ~ 1:ll(Ks) E9 U -+ l£(Ks d E9 Y so that U IS umt equivalent to U /, i. e., there is a umta1"Y operator U: X -+ 1£ Ks so that [~ ~1 [~ ~Y1 = [EB~OlU ~1l~ ~1· PROOF. Since U is a canonical functional-model colligation for S, the opm- tors A and B solve the Gleason problems (2.7) and (2.8 , respectively. By Prop<>- sition 2.4, this is equivalent to identities A*Zrow«)*Ks("()Y = Ks("()Y - Ks ·,0 Y B* Zrow«)*Ks(" ()y = S«)*y - S(O)*y. Besides, C and D are defined by formulas (2.12). Substituting their adjoints from (2.13) into the two latter equalities we arrive at (2.17) and (2.18. As v.e ha -e seen, equalities (2.17) and (2.18) imply (2.19) and (2.20). Equality 2.20 PIO'l'S statement (3). Equality (2.19) gives (2.22) V(I - A*Zrow«)*C·y = VKs(" ()y = ll(Ks). {EBd {EBd yEY !,lEY Thus the identity C(I - Zrow(z)A)-l/ == aleads to (J, (I - A*Zrow(Z)*)-lC'Y "" 0 for every z e IRd and y e Y; this together with equality (2.22) implies f == 0, and it follows that the pair (C, A) is observable. On the other hand, equalities (2.17) and (2.18) are equivalent to (2.16). Sub- stituting (2.19) into (2.16) and in (2.14) (for z = ( and y = y') gives U. rZrow«)*(1 - A*Zrow«)*)-lC*Y) = r(I-A*Zrow«)*)-lC*y1 l y l S«)*y and
- 28. TRANSFER-FUNCTION REALIZATION 11 respectively. The two latter equalities tell us that U· is isometric on the 'Du. (see (2.5» and therefore U is weakly coisometric. For the proof of part (4) we refer to [10, Theorem 3.4]. 0 Definition 2.5 does not require U to be a realization for S: representation (2.4) is automatic once the operators A, B, 0 and D are of the required form. We can look at this from a different point of view as follows. Let us say that A: 1I.(Ks) -+ 1I.(KS)d is a contractive solution of the Gleason problem (2.7) if in addition to (2.7), inequality d (2.23) LI(Af)kl~(Ks) ::; Ifll~(Ks) -lIf(O)II~ k=l holds for every f E 1i(Ks). It is readily seen tha.t inequality (2.23) can be equiva.- lently written in operator form as where the operator 0: 1i(Ks) -+ Y is given in (2.12). It therefore follows from Definition 2 5 that for every canonical functional-model colligation U = l~ g] for S, the operator A is a contractive solution of the Gleason problem (2.7). The following theorem provides a converse to this statement. Theorem 2.10. Let S E Sd(U,Y) be given and let us assume that 0, Dare !I' en by fonllulas (2.12). Then 1 For every contractive solution A of the Gleason problem (2.7) for 11.(Ks) , there exists an operator B : U -+ 11.(Ks) such that U = [~ g] is contrac- h e and S zs reahzed as in (2.4). 2 E ery such B solves the 1I.(Ks)-Gleason problem (2.8) so that U is a canonical functional-model colligation. PROOF. Smce A solves the Gleason problem (2.7) and since 0 is defined as in 2.12 , we conclude as in the proof of Theorem 2.9 that identity (2.17) holds which is equivalent to (2.19). On account of (2.19), it is readily seen that (2.18) and (2.20) are equivalent. But (2.20) is just the adjoint form of (2.4) whereas (2.18) coincides with 2.10 since D = S(O)) which in turn, is equivalent to (2.8) by Proposition 2.4. Thus, it remains to show that there exists an operator B*: 1I.(Ks) -+ U completely detew.ined on the subspace 'D C 1I.(Ks) by formula (2.10) and such that U· = [~: ~: 1is contractive. This demonstration can be found in [10, Theorem 2.4]. 0 3. de Branges - Rovnyak kernels associated with a Schur - Agler-class function on the polydisk Here we introduce a generalized Schur class, called Schur - Agler class, associ- ated with the unit polydisk Jl)d = {z = (Zl, ••• ,Zd) E Cd: Iz,.1 < 1 for k = 1, ... ,d}. We define the Schur-Agler class SAd(U,Y) to consist of holomorphic functions S: nd -+ £(U, Y) such that IS(T) II ::; 1 for any collection of d commuting opera.tors T = (Tlo"" Td) on a Hilbert space JC with liT,.II < 1 for each k = 1, ... , d where the operator SeT) is defined 88 in (2.1).
- 29. 12 J. A. BALL AND V. BOLOTNIKOV The following result appears in [1,2,14] and is another multivariable analogup of Theorem 1.1. The reader will notice that analogues of both (la) and lb fr m Theorem 1.1 are missing in this theorem. Theorem 3.1. Let S be a c'(U, Y) -valued function defined on Del. The follow- ing statements are equivalent: (3.1) (1) (c) S belongs to the class SAd(U, Y), i.e., S satisfies the von NeufTl.4Tlrt inequality IS(Tl. ... ,Td)1I :5 1 for any commutative d-tuple T ::: (T1, ••• , Td) of strict contraction operators on an a1.lXtl,ary Ht1hert space /C. (2) There exist positive kernels KI, ... ,Kd:]Dd x]Dd -+ c'(Y) such that/or every z =(Zl. ... , Zd) and ( = «(1. •.• , (d) in]Dd, d ly - S(z)S«()* = 2)1- z.(.)K.(z,(). k=l (3) There exist Hilbert spaces Xl,"" Xd and a unitary connectmg operator U of the structured form [ A B1 lA~l ... A~d ~11l~11 l~11 (3.2) U = 0 D = A:dl A: dd ~d : ;d -+ ;d (3.3) (3.4) 0 1 Od D U Y so that S(z) can be realized in the form S(z) = D +0(1 - Zdiag(Z)A) -1 Zdiag(z)B for all z E Dd where we have set (4) There exist Hilbert spaces Xl, ... I Xd and a contractive connecnng operator U of the form (3.2) so that S(z) can be realized in the form (3.3) Remark 3.2. Although statement (4) in Theorem 3.1 concerning contracti1l realizations does not appear in [1,2,14], its equivalence to statements (1)-(3) can be seen by anyone of the three approaches mentioned in Remark 2.2. Similar to the notion introduced above for the unit-ball case, there is a notion of weak coisometry for the polydisk setting as follows. Definition 3.3. The operator-block matrix U of the form (3.2) is weakly cois/)" metric if the restriction of U· to the subspace (3.5) Vu.:= V [Zdlag«()*(1 - A~Zdiag«()·)-lO*Y] c [~d] <eDd IIeY is isometric.
- 30. TRANSFER-FUNCTION REALIZATION 13 When U is given by (3.2) and S(z) is given by (3.3), it is immediate that we ha.ve the equality [C(1 - Zdia.g(z)A)-lZdia.g(Z) I} U = [C(1 - Zdiag(z)A)-l S(z)}. From this it is easy to verify the following general identity: (3.6) 1- S(z)S()" = C(1 - Z(z)A)-l(1 - Z(z)Z()")(I - A"Z(t)-1C" where here we set Z(z) = Zdia.g(Z) for short. It is readily seen from (3.6) that the wea.k-coisometry property of the colligation (3.2) is exactly what is needed to guarantee the representation (3.7) 1- S(z)S()" = C(1 - Zdia.g(z)A)-1(1 - Zdiag(Z)Zdia.g()")(I - A" Zdiag(),,)-1C". Note that the representation (3.7) has the form (3.1) if we take 3.8) Kk(z, () = C(1 - Zdiag(z)A)-1 Px"(I - A"Zdiag(),,)-1C" for k = 1, ..• , d, where Px" is the orthogonal projection of X := E9~=1 Xi onto Xk • 3.1. Weakly coisometric canonical functional-model colligations. Let us say that a collection of positive kernels {K1(z,(), ... ,Kd(Z,()} for which the decomposition 3.1) holds is an Agler decomposition for S. In view of (3.7), we see that a ' 3.3) for S arising from a weakly coisometric colligation matrix U 3.2 determines a particular Agler decomposition, namely that given by (3.8). Our next goal is to find a canonical weakly coisometric realization for S com- patible with the given Agler decomposition. Toward this goal we make the following definitions. Suppose that we are given a Schur-Agler class function S E SAd(U,y) to- gether with an Agler decomposition {K1(z, (), ... , Kd(Z, ()) for S. We set K(z, () = K 1(z, () +... +Kd(Z, (). Then lK is also a positive kernel on Jl))d and the associated reproducing kernel Hilbert space 1l K) can be characterized as d 1l(K) = L Ii : Ii E 11.(Ki) for i = 1, ... ,d i=1 with norm given by i=1 where 8: E9~-11l(K,) -+ 11.(K) is the linear map defined by (3.9) d where J = EBIi :=:: i=1 • • • •
- 31. 14 J. A. BALL AND V. BOLOTNIKOV It is clear that kers ={J E EB~=lll(Ki) : /l(z) +... + Jd(Z) == O}. If we let l Kl(Z,()] 11'(z, () ;= : ' Kd(z,() (3.10) we observe that by the reproducing kernel property, d (3.11) (I, 11'(., ()Y)E9t~ll£(K,) = L)Ii,K i (·, ()Y}l£(K,) i=1 so that (3.12) s·: 1K(·, ()y --+ 11'(., ()y. Furthermore, d (kers).L = V11'(.,()y c ffill(Kk ). (EDd k=1 yE)I We next introduce the subspace (3.13) v = VZdiag«)"''ll'(·,()y (ED" yE)I of EB~=11l(Kk) and observe that its orthogonal complement can be described as d d d V.L = {I = ~Ii E ~ll(K,) : ~Z.i,(z) =o}- In addition, the straightforward computation d 1111'(·,()YIla,~=l1t(Kk) = 2:(Kk«,()Y,Y}Y = (K«,()y,y}y = 1I1K(·,()y ~(K) k=1 combined with (3.12) shows that s* is an isometry, Le., that s is a coisometry· We remark that all the items introduced so far are uniquely determined from decom- position (3.1). Given an operator-block matrix A = [A'Jlt..,=1 acting on EB~=1 ll(K,), we;:} that A solves the structured Gleason problem lor the kernel collection {KlI"" if the identity d (3.14) !I(z) +... +Id(Z) - [/1(0) + ... + ideO)] = :Lz.(Af),(z) ,=1 d d md (Af).(Z) holds for all 1= EB.=1 I. E EBi =1 ll(Ki), where we write (AI)(z) == 1;17>==1 E EB~_lll(K.). Note that (3.14) can be written more compactly as d d (3.15) (sl)(z) - (sl)(O) = Lz,(A••f)(z) for all I E E9ll(K/r)' , 1 " 1
- 32. TRANSFER-FUNCTION REALIZATION 1~ where s is given in (3.9) and where d (3.16) A•• = (A.l ... A'd)! Et)1l(Kk) -41l(K.) (i = 1, ... ,d) k=l so that d d d (3.17) A=Et)A•• =lA'Jl~'J=l: Et)1l(Kk)-4Et)1l(Ki). k=l i=l We say that the operator B: U -4 Ea:-l1l(Kk) solves the structured ll(Kk)- Gleason problem for S if the identity (3.18) S(z)u - S(O)u = zl(Buh(z) +... +zd(Bu)d(Z) holds for all u E U. The following is the parallel to Proposition 2.4 for the polydisk setting. Proposition 3.4. The operator A: Eat=l1l(Ki ) -4 Eat=l1l(Ki) solves the stru tured Gleason problem (3.15) if and only if the adjoint operator A* has the folloWtng a ton on SpecIal kernel functions: (lKl·,()y K1(·,()y Kl("O)y • • • • • • - • • • for all ( E lIJ)d and y E y. The opemtor B. U -4 Ea~=lll(Ki) solves the structured Gleason problem (3.18) f S fad on y if the adjo~nt operator B*: Eat=lll(Ki ) -4 U has the following act n n spectal kernel junctions: ( Kl - ()y • • • (dKd("()Y t-+ S«()*y - S(O)*y for all ( E lIJ)d and y E y. PROOF. Making use of notation (3.10) and (3.4) we can write the definitions of A* BDd B* more compactly as 3.19 3.20 A*Zdiag«()*11'(·, ()y = 11'(., ()y - 11'(., O)y, B*Zdiag«()*11'(·, ()y = S«()*y - S(O)"y. By calculation (3.11), 1,1'(·,()y -1'(" O)Y)E9~=l 1i(K,) = (sf) «() - (sl)(O), y}y. On the other hand, it follows by the reproducing kernel property that I, A*Zd,ag(Z)*1'(·, Z)Y)E9~ 11i(K,) = (Zdiag(z)AI, 11'(., Z)Y)E9t~l1i(K,) d = "ElZdiag(z)A!1i(Z),y ~=l )I d = "EzilAf).(z), Y and the two latter equalities show that (3.15) holds if and only if (3.19) is in force for every y E y. Equivalence of (3.18) and (3.20) is verified quite similarly. 0
- 33. 16 J. A. BALL AND V. BOLOTNIKOV The following definition of a canonical functional-model colligation is the ~ logue of Definition 2.5 for the polydisk setting. Definition 3.5. Given S E SAd(U,Y), we shall say that the block-()peratr.t matrix U = [~g1of the form (3.2) is a canonical functional-model coli gatlon associated with the Agler decomposition (3.1) for S if (1) U is contractive and the state space equals ffi~=11l(K.). (2) A: EB1=11l(Ki) -4 ffi1=11l(Ki) solves the structured Gleason problem (3.15). (3) B: U -4 EB1=11l(Ki) solves the structured Gleason problem 3.18 for S (4) The operators C: EBt=11l(Ki) ~ Y and D: U ~ Y are given by (3.21) C: f(z) ~ (81)(0), D: 11. ~ S(O)1I.. Remark 3.6. For C and D defined in (3.21), the adjoint operators are gi."1!Ii by (3.22) C*: Y~ 'll'(',O)y D*: y ~ S(O)*y. The formula for D* is obvious while the formula for C* follows from equalities (I, C*Y)(Bt=l1i(K;) = (Cf, y)y = ((sl)(O), y}y = (f,'ll' ·,O)y (B =11l K holding for every f E EB1=11l(Ki). Theorem 3.7. Let S be a given function in the Schur-Agler class SA.:! U,y) and suppose that we are given an Agler decomposition (3.1) fOT S. Then there e:nsts a canonical functional-model colligation associated with {K1, .. " Kd}. PROOF. Let us represent a given Agler decomposition (3.1) in the inner product form as d l)(iKi(" ()y, ZiKi(', z)y')1i(K,) + (y, y')y ~l d = L (Ki (·, ()y, K.(., z)y'ht(K,> + (S«()*y, S(z)'!I~u, i=1 or equivalently, as (3.23) ([Zdiag(();'ll'("()y], [Zdiag(z);!(.,z)y']) (EB~=l 'H(K,))GlY 1['ll'("()Y) ['ll'("Z)y']) = S«()*y , S(z)*y' «(B~=l1l(K,))$U where l' is given in (3.10). The latter identity implies that the map (3.24) V: [Zdias(()*'ll'("()y] ~ ['ll'(" ()y] y S«()*y extends by linearity and continuity to an isometry from Vv =V EB Y (8. subspace of (EBt=lll(Ki )) EB Y-(3.13) for definition of V) onto 'R- = V ['ll'("()Y] C [ffi~_lll(Ki)] V S«()*y U ' ~EDd,IlEY
- 34. TRANSFER-FUNCTION REALIZATION - Let us extend V to a. contraction U· : (3.25) Computing the top and bottom components in (3.25) gives (3.26) (3.27) A·Zdie.g«).'11'(., ()y +C·y = '11'(" ()y, B· Zdie.g«)·'11'(·, ()y + D·y = S«ry. 11 Letting ( = 0 in the latter equalities yields (3.22) which means that C and D are ofthe requisite form (3.21). By substituting (3.22) into (3.26) and (3.27), we arrive at (3.19) and (3.20) which in turn are equivalent to (3.15) and (3.18), respectively. Thus, U meets all the requirements of Definition 3.5. 0 We have the following parallel of Remark 2.8 for the polydisk setting. Remark 3.B. As a consequence of the isometric property of the operator - V 3.24) introduced in the proof of Theorem 3.7, formulas (3.19) and (3.20) can be extended. by linearity and continuity to define uniquely determined operators AD: 'D -+ ffi~=lll(K.) and BD: 'D -+ U where the subspace 'D of ffi~=lll(Ki) is defined. in (3.13). In view of Proposition 3.4, we see that the existence question is then settled.: any opemtor A: ffi~=l1-£(Ki) -+ ffi~=lll(Ki) such that A· is an ofAD from 'D to all offfi~=lll(Ki) is a solution of the structured Gleason 3.15) and any opemtor B: U -+ ffi1=11-£(Ki ) so that B· is an extension of the operator BD: 'D -+ U is a solution of the structured Gleason problem (3.18) forS. In the polydisk setting we use the following definition of observability: given an operator A on ffi~=l.:ti and an operator C: ffi~=lXi -+ y, the pair (C,A) wi11 be called obsenJable if equalities C(1 - Zdiag(z)A)-lPx,x = 0 for all z in a neighborhood of the origin and for all i = 1, ... , d forces x = 0 in ffi~=l Xi. The latter is equivalent to the equality 3.28 V Px.(1 - A·Zdiag(Z)·)-lC·y = Xi for i = 1, ... , d zEa,I/E)/ for some neighborhood f:l. of the origin in Cd. The following theorem is the analogue of Theorem 1.2 for the polydisk setting; portions of this theorem appear already in {14, Section 3.3.1]. Theorem 3.9. Let S be a function in the Schur - Agler class SAd(U, Y) with a gtven Agler decomposition {K1 , •••• Kd} for S and let us suppose that (3.29) u a canonical functional-model colligation associated with this decomposition. Then: (I) U is weakly coisometric. (2) The pair (C. A) is obsenJable in the sense of (3.28). (3) We recover S(z) as S(z) = D +0(1 - Zdiag(z)A)-lZdle.g(z)B.
- 35. 18 J. A. BALL AND V. BOLOTNIKOV (4) Iff; = (~~1 :(EB~=l x.) ffiU ~ (EB~=l x.) ffi Y is any other co tg~ matrix enjoying properties (1), (2), (3) above, then there l8 Q ca~ functional-mo~l colligation U = [~Z 1as in (3.29) whICh l8 ~ equivalent to U in the sense that there are unitary operators U . X. -+ 1i(Ki) 80 that (3.30) [ A B) rEB~=l Ui 0) = rEB~=l U. 01 r~ ~l C D l 0 IJ1 l 0 lu lc Dr PROOF. Let U =[~ Z1be a canonical functional-model realization of S a&sI). ciated with a fixed Agler decomposition (3.1). Then combining equalities 319 (3.20) (equivalent to the given (3.15) and (3.18) by Proposition 3.4 and also for- mulas (3.22) (equivalent to the given (3.21)) leads us to (3.31) i(·, ()y = (1 - A"Zdiag«)*)-l'lI'(·,O)y = (1 - A"ZdIag ( " -lc;*y and (3.32) 8«()*y =8(0)*y +B"Zdiag«)*'lI'(·, ()y = D"y +B"ZdIag ( "T . C Substituting (3.31) into (3.32) and taking into account that y E Y is arbItrary get (3.33) 8«)* = 8(0)" +B" Zdiag«)"(1 - A"Zd. g ( " -le" which proves part (3) of the theorem. Also we have from 3.31 and 3.1 , V P'H.(K;)(l-A"Zdiag«)")-lC"y= VP'HK 1'.,(y <End (End yEJI yEll = VK, ., ( y = 11. K, (ElDd yEll and the pair (C, A) is observable in the sense of (3.28). On the other hand, equahoe5 (3.19), (3.20) are equivalent to (3.25). Substituting (3.31) into 3.25 and mto identity (3.23) (for z =( and y = y') gives U. [Z«)" (1 - A"Z«)")-lC"y) = r(1-A"Z«)")-1C* y1 y l 8«)"y and lZ«()"(I - A~Z«)")-lC"Y1 = ((1-A"~~~~~-lC"Y1 ' respectively. The two latter equalities show that U* is isometric on the space Vu- (see (3.5)) and therefore U is weakly coisometric. To prove part (4), let us assume that (3.34) 8(z) =8(0) +C(1 - Zdiag(z)A)-l Zdiag(z)B is a weakly coisometric realization of 8 with the state space EB~=l X, and such tha.t the pair (C,A) is observable in the sense of (3.28). Then 8 admits en }.gler decomposition (3.1) with kernels K, defined as in (3.8): K,(z, () =C(l - Zdiag(z)A)-l px•(1-A"Zdlag«)*) lC"
- 36. TRANSFER-FUNCTION REALIZATION 19 for i = 1, ... , d. Let ll(Ki) be the associated reproducing kernel Hilbert spaces and let :r. :X, -+ X = ffi~=l X, be the inclusion maps ~: Xi -+ 0 e ... e 0 e Xi e 0 e ... e o. - - Since the pair (C, A) is observable, the operators U,: X, -+ ll(K.) given by - - 1 (3.35) U,: Xi -+ C(l - Zdia.g(z)A)- Z.Xi are unitary. Let US define A E .c(ffi~=lll(K,)) and B E .c(U, ffit=11l(K.)) by (3.36) and ,=1 ,=1 .=1 In more detail: A = IA"l~"=1 where . - - -1 - - -1 - (3.37) A". C(l - Zdia.g(z)A) I,xj -+ C(l - Zdiag(z)A) ~AijXj. Define the operators A,. as in (3.16) and similarly the operators Ai. for i = 1, ... , d. Take the generic element f of ffi~=11l(Ki) and X E X in the form d 3.38 f z) = EBO(lx - Zdie.g(z)A)-IIjXj, 3=1 d X=EBXjEX. j=1 By 3 37 • we have 3.39 d • • • =LAij(O(l - Zdiag(z)A)-1IjXj) ;=1 d = L 0(1 - Zdiag(z)A)-I~AijXj 3=1 - .. 1 - = C(l - Zdie.g(z)A)- ~Ai.X, For f and :z: as in (3.38), we have d d sf) z) = L O(lx - Zdiag(z)A)-IIjxj = O(lx - Zdiag(z)A)-1 ~ IjX; ,=1 j=1 = O(1x - Zdiag(z)A)-l x which together with (3.39) gives (sf)(z) - (sf)(O) = 0(1- Zdia.g(z)A)-lx - Ox - - 1 .. = C(l - Zdie.g(z)A)- Zdiag(z)Ax d d = E zi .0(1- Zdie.g(z)A)-lI,Aj.x :::: L Zj • (Aj.f)(z), 1=1 j~l
- 37. 20 J. A. BALL AND V. BOLOTNIKOV which means (since f is the generic element of EB~=lll(K.)) that the operatms Ale,'" ,AM satisfy identity (3.15). Furthermore, on account of (3.38), (3.35 and (3.34), d d LZi(BuMz) =LZiC(l - Zdiag(z)A)-lI,Biu i=1 i=1 d =C(I - Zdiag(z)A)-l I>.I;B.u ..=1 - - 1 = C(l - Zdiag(z)A)- Zdiag(z)Bu = S(z)u - SOu and thus, B solves the Gleason problem (3.18) for S. On the other hand, for an:r of the form (3.38), for operators Ui defined in (3.35), and for the operator C defined on EBt=1'H.(Ki ) by formula (3.21), we have ( d d d d C ~Ui) X= tr(UiXi) (0) = ~C(l - Zdiag(O)A)-lI,x. = C~:r.z. =C:r and thus C(EBt=l Ui) = C. The latter equality together with definitions 336 implies (3.30). Thus the realization U = [~ Z1is unitarily equivalent to the ong· inal realization U = (~~1via the unitary operator EB~=l U.. This realization IS a canonical functional-model realization associated with the Agler decompootion {Kb ... , Kd} of S since all the requirements in Definition 3.5 are met. 0 We conclude this section with a theorem parallel to Theorem 2.9. In analogy with the ball setting, we say that the operator A on EB~=lll(K. is a controctn-e solution of the structured Gleason problem for the kernel collection {K1, ••• ,Kd} if in addition to identity (3.15) the inequality d IIAfll~t=l1i(Ki) := LIIAi.fll~(K.) ~ IIfll~~=l 1i(K.) - (sl)(O) ~ i=l holds for every function f e EB:=lll(Ki) or equivalently, the pair (C,A) is con- tractive: A'"A+C'"C ~ 1, where C: EBt=l 'H.(Ki) -4 Y is the operator given in (3.21). By Definition 3.5, {or every canonical functional-model colligation U = (~ ZJassociated with a gh-en Agler decomposition of S, the operator A is a contractive solution of the structured Glea.son problem (3.15). Theo1:em 3.10. Let t~.l) be a fixed Agler decomposition of a given juncnOfi S E SAd(U,y) and let C and D be defined as in (3.21). Then (1) For every contractive solution A of the structured Gleason problem (3.15), there is an operator B = EBB,: U -+ EB:=lll(K.) such that U =l~ Zl is a canonical functional-model colligation for S. (2) Every such B solves the Gleason problem (3.18) for S. PROOF. We start the proof with two preliminary steps.
- 38. TRANSFER-FUNCTION REALIZATION 21 Step 1. Let A of the form (3.17) solve the Gleason problem (3.15). Then d (3.40) 0(1 - Zdia.g(z)A)-1f = (81)(Z) Z E Jl)dj f E E91i(Ki) ,=1 where sand C are defined in (3.9) and (3.21), respectively. PROOF OF STEP 1. To show that identity (3.15) is equivalent to (3.40) we take A in the form (3.17) and define the operators Ale 0 0 0 A2• • ~ • • • (3.41) Ale = A2e = Ad. = • , • , • , ..., • • 0 • • 0 0 Ad. d SO that A•• : E9~=1 1i(K,) -+ E9~=11i(Ki) and A = 2:Ai.' On account of (3.41) .=1 and due to the block structure (3.4) of Zdia.g(Z) we have -1 .... ,. -1 (1 - Zdia.g(z)A) = (1 - ZlAh - '" - ZdAd.) Applying the operator e(l - Zdiag(z)A)-1 to an arbitrary f E E9~=11i(Ki) and making use of formula (3.21) for C, we get 3.42 e 1 - Zdtag(z)A)-lf 00 = e~)zlAh +... + zdAd.)kf k=O d d =(81)(0) +LZi(sAi.I)(O) + 2: zizj(sAi.Aj.I)(O) +.... •=1 i,j=l On the other ba,nd, by writing (3.15) in the form d (sl)(z) = (sl)(O) +L:zi(sAi.I)(Z) ;'=1 and iterating the latter fOlmula for each f E E9~=11i(Ki)' we get (3.43) (s/)(z) d d = (81)(0) + L Z;1 (SA31.1)(0) + L Zh (sAh.Ail.f)(O) +.,. ]1=1 h=l d + L Zjh [(sAj/o. '" A3~.Ajl.I)(0) +...J... . 310=1
- 39. 22 J. A. BALL AND V. BOLOTNlKOV Since the right-hand side expressions in (3.42) and (3.43) are identical, 3.40 f0l- lows. Now we have from (3.40) (I, (1 - A*Zdlag(Z)*)-lC*y) =(C(1 - Zdiag(Z)A)-1f, y) = «sf) z ,y = /,T·,zy for every z E lDid and f E EBt=l'H.(Ki ), and thus, equality (3.31) holds. 0 Step 2. Given operators A, C and D with A 0/ the form 3.17) equal to contractive solution of the structured Gleason problem for the kernel collechon {Kl, ... ,Kd} and with C and D given by (3.21), ifU = (~Zl t.s a ~e realization of S for some operator B =EB B,: U ~ ffi~=1 'H.(K.), then B salta jh, EB~=l1i(Kk)-Gleason problem for S, i.e., B satisfies identity 3.18. PROOF OF STEP 2. Since U = [~Z] is a realization for S, equalrty 3.33 holds. Making use of equality (3.31) (which holds by Step lone can write 3.33 as B*Zdiag«)*'lI.'(·, ()y + D*y = S ( .y or, in view of formula (3.21) for D, as (3.44) B* Zdiag«r'lI.'(·, ()y = S«)*y - S 0 .y. Taking the inner product of both parts in (3.44) with an arbitrary funw n JE EBt=l1i(Ki) leads us to Zdiag(z)Bu = S(z)u-S(O u which is the same as 3.18 0 To complete the proof of the theorem, it suffices to show that there exists an operator B : U -+ EBt=l 1i(Ki) such that equality (3.33 holds for every u EU and the operator matrix (3.45) U* = [A* C*1. rffit=1 'H.(K·)1 ~ rffi~=1 'H. K )1 B* D* . l y l u is a contraction. As we have seen, equality (3.33) is equivalent to (3.44 , v-htch in turn, defines B* on the space 'D introduced in (3.13). Let us define B: 1> -i EB:=l1i(Ki ) by the formula B: Z«)*'lI.'(.,()y = S«)*y - S(O)·y and subsequent extension by linearity and continuity; it is a consequence of the isometric property of the operator V in (3.24) that the extension is well-defined and bounded. We arrive at the following contractive matrix-completion problero: d - find B: U -+ EBi=l 1l(K,) such that B*!'D = B and such that U· of the form (3.45 is a contraction. Following [10] we convert this problem to a standard matrix- completion problem as follows. Define operators d d Tn: Vi -+ E91i(K,), T12: 'D ~y ~ E9ll(K,), .-1 =1 by (3.46)
- 40. TRANSFER-FUNCTION REALIZATION 23 Identifying we then can represent U· from (3.45) as (3.47) 1>.1. -+ EB~=l 1i(K,) 1>Ef)Y U where X = B*v.L is unknown. Thus, an operator B gives rise to a canonical functional-model realization U = t~ g1of S if and only if it is of the form X. 'D.1. It B = B* :U -+ 'D ~ E!11i(K,) ,=1 where X is any solution of the contractive matrix-completion problem (3.47). But this is a standard matrix-completion problem which can be handled by the well- known Parrott's result [291: it has a solution X if and only if the obvious necessary conditions hold: 3.48 :Making use of the definitions of Tn. T12,T22 from (3.46), we get more explicitly Tt2 A"v C" [Tu T121 = [A* Co01. T22 = B D'" Thus the first in (3.48) is contractive since A is a contractive solution of the stru tured Gleason problem (3.15), while the second expression collapses to - V see f rmula 3.24) which is isometric by (3.1). We conclude that the necessary nditi ns 3.48 are satisfied and hence, by the result of [29], there exists a solution X to problem 3.47. This completes the proof of the theorem. 0 4. de Branges - Rovnyak kernels associated with a Schur- Agler-class function on a domain with matrix-polynomial defining function A generalized Schur class containing all those discussed in the previous sections as special was introduced and studied in [4,9] (see also [5] for the scalar-valued and can be defined as follows. Let Q be a p x q matrix-valued polynomial 4.1 such that 4.2 Q(z) = qu(z) .. . q1q(Z) • , , • • • Q(O) = 0 and let 'DQ E en be the domain defined by 'DQ = {z E en: Q(z)1I < I}. Now we recall the Schur Agler class SAQ(U, Y) that consists, by definition, of £. U, Y)-valued functions S(z) = S(Zl" '" zn) analytic on 'DQ and such that IIS(T) II ::; 1 for any collection of n commuting operators T = (T1, ... ,Tn) on a. Hilbert space K:, subject to Q(T)U < 1. By [5, Lemma 1], the Taylor joint spectrum of the commuting n-tuple T = (T1,.•. ,Tn) is contained in 'DQ whenever IIQ(T)II < 1, and hence SeT) is well defined by the Taylor functional calculus (see [19]) for any £. U, Y)-valued function S which is analytic on 'DQ. Upon using K: = e and T, = z,
- 41. J. A. BALL AND V BOLOTNIKOV for j 1, ... ,n where (Zh"" zn) is a point in VQ we conclude that any fun<'t SAQ(U,Y) is contractive-valued, and thus, the class SAQ U,Y) is th Ia the Schur class SPQ (U, Y) of contractive valued functions analyt'c on 'DQ B von Neumann result, in the case when Q(z) =z, these classes coinClde- in getll!r:!l. SAQ(U, Y) is a proper subclass of SPQ(U, Y). The following result appears (see also [5] for the scalar-valued case U - Y = C) and is yet another m b analogue of Theorem 1.1. We will often abuse notation and will wnte Q z lIlStead of Q(z) ® 1 where 1 is the identity operator on an appropnate Hilbert space from the context. When the following theorem is viewed as a parallel fTh rem we see that, just as in the polydisk setting, there is no parallel to rut lIS and (lb). Theorem 4.1. Let S be a £(U, Y)-valued fund on defined on'DQ The lowing statements are equivalent: (4.3) (1) (c) S belongs to SAQ(U,y), (2) There exists a positive kernel ( 1K~1 .. . lK~p1 IK =: : : 'DQ 1Kpl IKpp which provides a Q-Agler decomposition f r S .e, su that f z,( E'DQ, P q (4.4) Iy - S(z)S«()* = L IKkk (Z, () - L L q z QT~ z ( k=1 k-l,=1 (2') There exist an auxiliary Hilbert space X a d a fu t& (4.5) H(z) = [Hl(Z) '" Hp(z] analytic on'DQ with values In .c(XP,)7) so that f revery z, ( E 'DQ (4.6) Iy - S(z)S«()* =H(z)(Ix - Q z Q ( * H ( *. (4.7) (3) There exist an auxiliary Hilbert space X and a u ~ tary ronnec g perot U of the form u = [~ ~]: [~] ~ [~] so that S(z) can be realized in the form (4.8) S(z) - D +C(Ixp - Q(z)A) lQ(z)B for all Z E'DQ. (4) There exist an auxiliary Hilbert space X and a contractive connecun9 r erator U of the form (4.7) so that S(z) can be realized in the form (4. Remark 4.2. If S = [~!~ ~~~] E SAQ(U1$U2,)71 $Y2), then the block~t~ 5'J belongs to the Schur Agler class SAQ(U"YJ ) for i,j = 1,2. For the pro ' suffices to note that IIS'J(T)II $ IIS(T).
- 42. TRANSFER-FUNCTION REALIZATION Remark 4.3. The equivalence (2) .: :. (2') can be seen by using the Kol- mogorov decomposition for the positive kernel oc: (4.9) OC(z, () = • • • Hp(z) The implication (4) > (1) can be handled by any of the three approaches sketched in Remark 2.2. Following the approach from (10), we first handle the case where U is coisometric, using the identity (4.10) 1- S(z)S«()* = C(I - Q(z)A) 1(1 - Q(z)Q«()*) (I - A·Q«()*)-lC· • holding for S of the form (4.8) and U given by (4.7), the straightforward verification of which is based on the identity [CeI - Q(Z)A)-lQ(Z) I) U = (C(I - Q(Z)A)-l S(z)). Then the general (contractive) case follows by extension arguments and Remark 4.2. Remark 4.4. With no assumptions on the polynomial matrix Q(z) some de- gener ies ccur which ca.n be eliminated with proper normalizations. We note first fall th t it is natural to assume that no row of Q(z) vanishes identically; other- - wise one can cross out any vanishing column to get a new matrix polynomial Q(z) f smaller size which defines the same domain VQ in en. Secondly, in the second term of the Q-Agler decomposition (4.4), the (i, l)-entry lK.;,1 of OC is irrelevant for any pair of indices i,l such that at least one of qik(Z) and qlk(Z) vanish identically f r each k = 1, ... ,q. Note that if the first reduction has been carried out, then all wag nal entries oc.. are relevant in the second term of (4.4) in this sense. It f llows that, without loss of generality, we may assume that OCil(z,() == 0 for each such pair of indices (i, l). To organize the bookkeeping, we may multiply Q(z) on the left and right by a permutation matrices II and II' (of respective sizes P x P and q x q so that Q z) = IIQ(z)II' has a block diagonal form Q(l)(z) 0 - 4.11 Q(z) = • • • o Q(d)(z) with the ath block Q a) (0: = 1, ... ,d) of say size POI x qOl and of the form Q(a) (z) = [q(OI) (z))P.:: ~~ ',3 .-13-1 - and irreducible in the sense that Q has no finer block-diagonal decomposition after perll!utation equivalence, i.e., for each 0: for which Q(a) is nonzero and for any pair of indices i, l (1 :5 i, I :5 POI)' there is some k (1 :5 k :5 qj) so that either q.~ (z) or 'lJ~)(z) does not vanish identically. Without loss of generality we may - assume that the original matrix polynOInial Q is normalized so that Q == Q. We may then aSS11me that the positive kernel in (4.3) and (4.4) has the block diagonal
- 43. ~6 J. A. BALL AND V. BOLOTNIKOV decomposition (4.12) l lK(I) (Z, () lK(z,() = o where lK(a) in turn has the form l lK(a) 11 lK(a) = : lK(a) p",l Under the normalizing assumption that lK has this block diagonal f rm -1.12 Q(z) is written as a direct sum of irreducible pieces (4.11 ,the nstruct1 us to follow can be done with more efficient labeling but at the cost of an additI nal 1lf of notation. We therefore shall assume in the sequel that this diag nal structure • not been taken into account (or that the matrix polynomial Q is already irredu e until the very end of the paper where we explain how the polydisk settIDa can seen as an instance of the general setting. As in the previous particular settings of the ball and of the polydIsk, we mtro- duce the weak-coisometry property as the property equivalent to 4.10 llapsmo to 1 - S(z)S«)* = e(1 - Q(z)A)-I(I - Q(z)Q (* 1- A*Q ( • -Ie-. Definition 4.5. The operator-block matrix U of the form 4.7 is weakl lSO- metric if the restriction of U" to the subspace (4.13) .- V lQ«)"(1 - A*Q«)* -lc·y1 (Xq 1 'Du··- c y (EVQ Y yEY is isometric. Due to assumption (4.2), the space 'Du. splits in the form 'Du. ='DEBYwhere (4.14) 'D = V Q«)*(I - A*Q«)*)-IC"y c Xq • (EVQ,yEY 4.1. Weakly coisometric canonical functional-model Q-realizations. Let us suppose that we are given a function S in the Schur Agler class SAQ u,Y together with an Agler decomposition lK. as in (4.3) (so (4.4) is satisfied). We'9l,u use the notation Q.k«) for the k-th column of the polynomial matrix Q. What actually comes up often is the transpose: (4.15) Q.k«)T = (qlk«) q2k«) ... ~k«)]' Note that with this notation the Q-Agler decomposition for S (4.4) can be written more compactly as p q (4.16) 1:v - S(z)S«)* =L KIt,k(Z, () - L QJ)(z)lK(z, ()QJJ«)" It I j 1 an expression more suggestive of the Agler decomposition (3.1) for the pol)~ case.
- 44. TRANSFER-FUNCTION REALIZATION 27 We say that the operator A: ll(K)P ~ 1l(K)9 solves the Q-coupled Gleollon problem for 1-£(K) if P q (4.17) Z)!k,k(Z) - !k,k(O)) = L Q.k(z)T[A!1k(Z) for all I E ll(K)P 10=1 10=1 so each f E ll(K)P has the form 1= It • • • where 110 = • • • !k,P E ll(K). Similarly, we say that the operator B: U ~ ll(K)q solves the Q-coupled ll(K)- Gleason problem for S if the identity 9 (4.18 S(z)u - S(O)u =L Q.k(z)T[Bu)k(Z) holds for all u E U. 10=1 The following proposition gives the reformulation of Gleason-problem solutions in terms of the adjoint operators. In what follows, we let {ell .. " ep } to be the standard basis for CP. Proposition 4.6. The operator A: ll(K)P -+ ll(K)q solves the Q-coupled Gleason problem 4.17) if and only if the adjoint A* of A has the following ac- non on specrol kernel functwns: 4.19 • • • • • • - fo all (E VQ and y E y, where Ei = Iy ® e, for i = 1, ... ,p: Iy 0 o Iy 4.20 • • • o • • • o Iy • The opemtor B: U -+ 1l(K)9 solves the Q-coupled ll(K) -Gleason problem (4.18) for S 'if and only if B*: ll(K)q -+ U has the following action on special kernel functwns: (4.21 B*: • • • H S«)"'y - S(O)*y for all CE VQ and y E y. K(., ()Q.q«)T*y PROOF. We start with the identity K(z,()E1y (4.22) Q«)"' • • • - - • • • which holds for all z, ( E VQ and y E Yj once Q«)* is interpreted as Q«)*®Iy and similarly for Q.,.«)T*, this can be seen as a direct consequence of the definitions
- 45. 28 J. A. BALL AND V. BOLOTNIKOV (4.1), (4.3), (4.15) and (4.20). Letting (4.23) 1l'(z, () := : llK(Z' ~)E1J lK(z, ()Ep for short, we then can write formulas (4.19), (4.21) more compactly as (4.24) A*Q«)*1l'(·, ()y =1l'(', ()y -1l'(" O)y, (4.25) B*Q«)"1l'(·, ()y =8«)*y - 8(0)*y where now Q«)* is to be interpreted as Q«)" ® 11£(lK)' In the following computa- tions, Q«)* is either Q«)" ® Iy or Q«)" ® 11£(K) according to the context. B) the reproducing kernel property, we have for every f = EI):=1 I" E 1llK. P, p p (4.26) (I,'lr(',()Y)l£(IK)P = LU",lK(., ()E"y)l£(K) = I: Ei.f,,« ,y y "=1 "=1 Therefore, (4.27) (I,1l'(·,()y-1l'(·,O)Y)l£(IK)P = <t;u",,,«)-f,,,,, 0 ),Y)y' On the other hand, it follows again from (4.26) that (I,A*Q(z)*1l'("Z)Y)l£(IK)P = (Q(z)AI,1l'("Z)Y)l£(K)P = <t,[Q(Z)AllJJ Z,Y y and since p p q (4.28) ~[Q(z)Alkj(z) = L L <l.i"(z)[AJ],,,](z) j=1 j=I,,=1 we get (I, A*Q(z)"1l'(·, z)y)1£(lK)p = < t Q.,,(z)T[Afl,,(z), Y) y' "=1 Y the Since the last equality and (4.27) hold for every f E 1l(K)P, ( E VQ and y ~val~ce equivalence of (4.17) and (4.24) (which is the same as (4.19)) folloWS. EqUl of (4.18) and (4.25) follows by the same argument from equalities (1£, S«)"y - S(O)"y)u = (8«)1£ - 8(0)1£, y}y and (1£, B*Q(z)*1l'(·, z)y)u =(Q(z)Bu, 1l'(', Z)Y)l£(IK)P =(t[Q(z) Bulj,j (z),y) = (:tQ.,(z)T[Bti1.(M), 3=1 y "==1 Cl holding for all 1£ E U and y E :V.
- 46. TRANSFER-FUNCTION REALIZATION Just as in the particular cases discussed in the previous sections, it turns out that the formulas (4.19) and (4.21) can be extended by linearity and continuity to define uniquely determined bounded well-defined operators B;: V-+U as a consequence of the isometric property of the operator V defined below in (4.33). Definition 4.7. We say that the operator-block matrix U = l~ ~) :1i(OC)P EB U -+ 1i(JK)'l EB Y is a canonical functional-model colligation matrix for the given function S and Agler decomposition JK if 4.29 1) U is contractive. 2) The operator A solves the Q-coupled Gleason problem (4.17) for 1i(OC). 3) The operator B solves the Q-coupled 1i(OC)-Gleason problem (4.18) for S. 4 The operators C: 1i(OC)P -+ Y and D: U -+ Yare given by C: l1(z) • • • t-+ il,l(O) +... +fp,p(O), D: u t-+ S(O)u. F flnulas 4.25) can be written equivalently in terms of adjoint operators as follows: 4.3 C*: y t-+ 1'(" O)y D* : y t-+ S(O)*y where T is defined. in (4.23). The next theorem is the analogue of Theorem 3.7. Theorem 4.8. Let S be a given function in the Schur - Agler class SAQ(U, Y) and suppose that we are given an Agler decomposition (4.4) for S. Then there exists a canonical functwnal-model colligation associated with the kernellK.. PROOF. Let us rearrange the given Agler decomposition (4.4) or (4.16) as q p Iy +L Q.k(z)TOC(z, ()Q.k«)T* = S(z)S«)* +L EjOC(z, ()Ej, 10=1 j=l and then invoke the reproducing kernel property to rewrite the latter identity in the inner product form as q 4.31) ~(lK.(., ()Q.k«)T..y, lK.(', Z)Q.k(Z)T*y'ht(K) +(y, y'}y 10=1 p = ~(lK.(., ()EjY, OC(" z)E,Y')1{(K) +(S«)*y, S(z)*y')u. 3-1
- 47. 30 J. A. BALL AND V. BOLOTNIKOV The latter can be written in the matrix form as (4.32) / rQ«()*1l'(·, ()Y1 ' rQ(Z)*1l',(.,Z)Y'1) L Y L Y 'H(K)geY ( fT ("( y1 fT ',Z y'1 = ls (ty ,lS Z *y' The latter identity implies that the formula (4.33) V: rQ«)*1l'("()Y1 r-+ rT "()y1 L y lS ( .y extends by continuity to define the isometry from 1 = V ffi Y c 'Hex q Ee Y see (4.14) for definition of 'D) onto nv = V l~~(f:~1 c l1l~)p1· (E'DQ lIEY Let us extend V to a contraction U*: ['Hc;)9 1~ ['H~)P]. Thus, (4.34) U* - [A* c*1· fQ«)*T(., ()y1 r-+ fT("()Y1 - B* D* . l y lS«)*y • Computation of the top and bottom components in (3.25) gives (4.35) A*Q«)*1l'(., ()y +C*y = T(·, ()y, (4.36) B*Q«)*T(·, ()y + D*y = S«)*y. Letting ( =0 in the latter equalities and taking into account (4.2) leads ~tOt~4:Oof from which we see that C and D are of the requisite form (4.29). Substl tu 10uiv- (4.30) into (4.35) and (4.36) then leads us to (4.24) and (4.25) w~ch a:n:ucal alent to (4.17) and (4.18), respectively. Thus we conclude that U 18 a 0 functional-model colligation as wanted. . an operatol' For this general setting we define observability as follows: gI~n called Q. A: XI' -+ xq and an operator C: XI' ~ Y, the pair (C, A) will. be b hOod of observable if the identities C(I - Q(Z)A)-II.;x = 0 for all z in a neigh or denote the origin and for all i = 1, ... ,p forces x = 0 in X. By Z. : X -+ X" we
- 48. TRANSFER-FUNCTION REALIZATION a1 the inclusion ma.p which embeds X into the i-th component of XI' = X ffi .•. ~ X. Thus 0 Xl • • • • • • (4.37) ~: x,t-+ X, and r· • • X, t-+ X•. • • • • • • 0 xI' The Q-observability can be equivalently defined in terms of adjoint operators as (4.38 VrI;(I A*Q(z)*) lC*y:zEb.,yEY,i=1, ... ,p}=X where b. is some neighborhood of the origin in en. The following theorem is the analogue of Theorem 1.2 for the present general setting. Theorem 4.9. Let S be a function in the Schur-Agler class SAQ(U, Y), let the poSuve kernellK. of the foun (4.3) provide an Agler decomposition (4.4) for S a d suppose that U = t~ Z) :ll(lK.)p ffi U -t ll(lK.)q ffi Y is a canonical functional- model lltgatton assoctated with 8 and lK.. Then the following hold: 1 U 1.8 eakly C01.8ometnc. 2 The p (C,A) 1.8 Q-observable in the sense of (4.38). 439 3 "' reco er S as 8(z) = D +C(I - Q(z)A)-lQ(z)B. 4 If U = ~ ~ : XP ffi U -t xq ffi Y is another colligation matrix enjoying p ert es (1), (2), (3) above, then there is a canonical functional-model - col 'g tum U for (8, lK.) such that U and U are unitarily equivalent in the sense that there is a unitary operator U: X -t 1l(lK.) so that A B C D EI1~=l U o o _ EI1?=l U 0 Iy - 0 Iu - - A B CD' PROOF. Let U = t~ Z) be a. canonical functional-model realization of 8 asso- ciated with a fixed Agler decomposition (4.4). Then combining equalities (4.24), 425 equivalent to the given (4.17) and (4.18) by Proposition 4.6) and also for- mulas 4.30 equivalent to the given (4.29)) gives 4.40 1'("( y = (I - A*Q(()*)-l']['(., O)y = (I - A*Q(()*)-lC*y and 4.41 S(()*y = 8(0)*y +B*Q(()*']['(', ()y = D*y +B*Q(()*,][,(., ()y. Substituting (4.40) into (4.41) and taking into account that y E Y is arbitrary, we get 4.42) 8(()* = S(O)* + B*Q(()*(I - A*Q(()*)-lC· which proves part (3) of the theorem. Also we have from (4.40) V r.(I - A*Q(()*)-lC·y = V :r;']['(.,()y C;E"DQ IIEY. t-l,....,p CE"DQ,IIEY, t 1,..'IP
- 49. 32 J. A. BALL AND V BOLOTNIKOV and we can proceed due to (4.37) and (4.23) as follows: V X;1l'(.,()y= V JK(·,()E.V= V K(',()y=1£X C;EDQ,lIEY, i-I,...,,, C;E'DQ,lIEY, 1.=1,... ,1' <E'DQ yEY Thus the pair (C, A) is Q-observable in the sense of (4.38). On the other ban equalities (4.24), (4.25) are equivalent to (4.34). Substituting (4.40 into 4.34 an into identity (4.32) (for z =( and V =V') gives • [Q()'(I - A*Q()*)-IC*v1_ r(1- A*Q ( • -1C*V} u y - l S <'y and respectively. The two latter equalities show that U· is isometric on the space Vu (see (4.13)) and therefore U is weakly coisometric. To prove part (4), let us assume that (4.43) 8(z) = 8(0) +O(I - Q(z)A)-IQ(z B is a weakly coisometric realization of 8 with the state space X and such that the pair (0, A) is Q-observable in the sense of (4.38). Then 1- S(z)S()* = O(I - Q(z)A)-l(I - Q(z)Q (t 1- A*Q ( * -Ie- which means that 8 admits a representation (4.6) with H z = 0 1- Q z A- Let I; be given as in (4.37). Representing H in the form 4.5 with (4.44) we then conclude from Remark 4.3 that 8 admits the Agler decompositi n 44 with for i,j = 1, ... ,p. Let 1£(IK) be the reproducing kernel Hilbert space associated with the posithe kernellK =[1K'31~,j=I' Let us arrange the functions (4.44) as follows (4.45) ( HI.(Z)] lo(I-Q(.Z)A)_lXl] G(z) := : = : . Hp(z) O(I - Q(z)A)-IIp Since by construction lK(z,() = G(z)G()* and since (0,...1) is Q-observable, the formula (4.46) U: x ~ G(z)x defines a unitary map from X onto 1£(IK). Let us define the operators A: ll(iW-+ 1£(IK)q and B: U -+ l£(lK)q by (4.47) and B= (a,U)B. , 1
- 50. TRANSFER-FUNCTION REALIZATION In more detail, using representations - - - All ••• All' Bl - - A= • - : XI' ~ xq and B= • • • • • • • - - - Aql ••• Aqp Bq we define Au • •• All' and B= A= • • • • • • Aql ••• Aqp block-entrywise by 4.48) and • • • : U -i- xq, 33 for i = 1,•.. , q and j = 1, ... ,p. We next show that the operators A and B sol~e the Gleason problems (4.17) and (4.18), respectively. To this end, take the genenc element J of 1£(K)P in the form 4.49 J(z) = • • • where x:= • • • On account of 4.45), we have for f and x as in (4.49), P I' L:lk.k(Z) =L:O(l - Q(z)A)-lIkXk k=l k=1 P = C(I - Q(z)A)-lL:IkXk = O(l - Q(Z)A)-lx. k=l Therefore, aud since Q(O) = 0, we have p 4.50 L:(Ik.k(Z) - A,k(O)) = O(I - Q(z)A)-lx - Ox k=1 On the other hand, we have by (4.45) and (4.48), P {AflkJ(Z) = L:Ak,G(z)x, .=1 j p p = G(z) LAkiX, == O(l - Q(Z)A)-lIj L:AkiXi ,==1 j i=1 and it follows directly from (4.37) and (4.1) that P q P L z::'l.7k(z)I, L AkiXt == Q(z)Ax. 3=1 k=l s=1
- 51. 34 J. A. BALL AND V. BOLOTNIKOV Making use of the two last equalities and of (4.28) we get q P q L Q.k(Z)T[Aflk(z) = L:L:<l.ik(z)[AJ1k,j(Z) k=l j=lk=l P q P =L:L:<l.ik(z)0(1 - Q(Z)A)-lI, L:Ak,x. j=l k=l ,=1 P q P = O(I - Q(Z)A)-1 LL<l.7k(Z)I, L:Ak,x, j=l k==l 0=1 == 0(1 - Q(z)A)-lQ(z)Ax which together with (4.50) implies (4.17). Similarly we conclude from (4.15), (4.45 • (4.47) and (4.43) that q P q LQ.k(Z)T[Bu]k(Z) = LL<l.ik(z)[Bku],(z) k=l j=lk=l P q = L L <l.ik(z) [G(Z)BkU]j j=lk=1 P q = L L <l.ik(Z)O(I - Q(Z)A)-lI,Bku j=lk=l P q = 0(1 - Q(Z)A)-l L L <l.7k(z)I,Bku j=lk=l = 0(1 - Q(z)A)-lQ(z)Bu = S(z)u - S(O)u and thus, B solves the Gleason problem (4.18) for S. Finally, for f and:c of the form (4.49), for the operator U defined in (4.46) and for the operator C defined on 1l(lK)P by formula (4.29), we have P P P P C(EBU)x = Cf =Lik,k(O) = L O(I - Q(O)A)-lIkXk =LOIkXk =ex, i=l k=l k=l k=l and thus, C(~U) =0. The latter equality together with (4.47) implies (4.39). According to Definition 4.7, the colligation U = [~ g] is a canonical functional-model colligation associated with the Agler decomposition lK of S. 0 Let us say that the operator A: 1l(lK)P --+ 1l(IK)q is a contractive solution of the Q-coupled Gleason problem for 1l(1K) if in addition to identity (4.17) the inequality k IAfll~(K)4 ~ IIfl~(K)P - Llfk,k(O)II~ , 1 holds for every function f e 1l(lK)P or equivalently, if the pair (C, A) is contractive where C: 1l(lK)P ~ Y is the operator given in (4.29).
- 52. TRANSFER-FUNCTION REALIZATION 36 Theorem 4.10. Let (4.4) be tl fixed Agler decomposition of tl given function S E SAQ(U, Y) and let 0 and D be defined as in (4.29). Then (1) For every contractive solution A of the Q-coupled Gleason problem (4.17), there is an operator B: U ~ 1i(lK)q such that U = l~ ~1is a canonical functional-model colligation for S. (2) Every such B solves the Gleason problem (4.18) for S. The proof is very much similar to the proof of Theorem 3.10 and will be omitted. In conclusion, we compare functional model Q-realizations obtained in this section with particular cases considered in Sections 2 and 3. The unit ball setting. In this case, Q = Zrow (in particular, p = 1) and def- inition 2.2) can be interpreted as the (uniquely determined) Agler decomposition of the form (4.4) with the kernellK= Ks. Then (4.23) gives 'll'(z,() = Ks(z,() and 4.33 coincides with (2.15). Since all canonical functional-model colligations are obt.ained via contractive extensions of isometries V (from (2.15) for the unit ball setting or from (4.33) for the general Q-setting), it follows that realizations con- structed in Section 2 ca,D be obtained from those in Section 4 by letting Q = ZIOW' Moreover, if Q = Zrow, then observability in the sense of (4.38) collapses to ob- servability defined in part (2) of Theorem 2.9. The unit polydisk setting. In this case, Q = Zdiag, P = q = d, and the Agler representation (3.1) for an S E SAd(U,y) can be written in the form (4.4) Kl 0 with the kernel ][{ = • • • 4.51 . Then (4.23) takes the form 'll'(Z, () = • • • Kd(Z, () I8l ed where {eh--' ,ed} is the standard basis for ICd. Observe that (4.51) is not the same as 3.10 . Now 4.33) collapses to (lKl(Z, () I8l el Kl(Z, () I8l el 4.52) V: • • • (dKd(Z, () I8l ed y whereas 3.24) can be written as (lK1(z,() • - 4.53 V: , • (dKd(Z, () y , • • Kd(Z, () I8l ed S«()"y K1(z, () • , , Kd(Z, () S«()*y - To get canonical functional-model realizations as in Definition 3.5, we extend V to . - A" C" d ( d a contractIOn U" = B" D" : EB.=l1i Ki ) EB Y ~ EBi=l1i(Ki) EB u. If for such a contraction we let U to be of the form (3.2) with - - A., = A., I8l e.e;, OJ = OJ I8l ej,
- 53. 36 J. A. BALL AND V. BOLOTNIKOV then U· will be a contraction from E9~=1 (l£(K.))P ffi Y to EB~=1(1£(K.))P 9 U ex- tending the isometry V given in (4.53). It is not hard to see that U is a canonical functional-model Q-realization for S in the sense of Definition 4.7. Thus, any "poly- disk" canonical functional-model realization gives rise to a canonical functional- model Q-realization for S. Of course, the converse is not true. To see the polydisk setting as a particular instance of the general Q-setting we need to make use of the block-diagonal decomposition of Q into irreducib e parts discussed in Remark 4.4. For the polydisk setting with Q z) = ZdIag Z , this diagonal structure is nontrivial and already apparent. Thus we assume that Q(z) has the form (4.11) and the positive kernel OC giving rise to the Q-Agler decomposition (4.4) has the compatible block decomposition (4.12 . The Q-Agler decomposition (4.4) now has the form (4.54) I - S(,)S«), ~t.(E JKl-:>(',() - t 1:,i: (,)q,: ()K,;' z,(1 and can be rewritten in inner-product form as d qc> L L (oc(a) (', ()Q~~) «()T.y,oc(a) (', Z)Q.k(Z)T" y')ll(K a=1k=1 d Pc> = L L(lK.(a) (', ()E3 a )y, oc(a) (', z)Eja)Y'}ll(K" ) + s (ry, S Z 'Y1U a=1j=1 where E3a ) = Iy ® ej and where {el, ... ,ep,J is the standard basis for 0'... Then the isometry V in (4.33) has the form ( K(Q) (%.{)Ei Q ) 1 where T(a) (z, () = : and where V has domain equal to Dv :=: 'D al Y K(a) (%.{)E~:> where Rv = V l1t(:~'()·1 c 41£(oc(a)PQ EBUi cevQ.lleY T( )(',C)y 0-1 S«()*y all this specializes to (4.53) for the polydisk case. We say that the operator A: EB~ 11£(lK(a)PQ -+ E9: 11£(lK(a)qQ solves the Q-structuTed Gleason problem
- 54. TRANSFER.FUNCTION REALIZATION for the kernel collection {K(I), •.• ,K(d)} if for all f = d Pa d qa L L(J~~2(z) - f~~2(0)) =1:L Q~~)(z)TlAf1~a)(z) a=lk=1 a-lk 1 1(1) • • • • • • ' (a) I(a) p" k,p", 37 We say that the operator B: U -+ E9!=lll(K(a»)q", solves the Q-structured {K 1), .... K d)}-Gleason problem lor S if d qa S(z)u - S(O)u = L L Q~~) (z)[Bu1~a) (z) a=1 k=l for all u E U. where we write Bu = [Bult • • • [Bu](a) k,l {Bu1 a = : where in turn (Bul~a) = : E ll(K(a»). We de- • • [Bu] " [Bu](a) 0: k,po fine a canonical functional-model colligation matrix U for a given function S E SAq U.Y and left Q-Agler decomposition {K1, ... ,Kd} (so (4.54) holds) to be any perator-matrix U = {~gl: E9!=lll(K(a»)Pa ffiU -+ E9!=lll(K(a»)qa ffiY so that 1 U is a contraction. 2 the operator A solves the Q-structured Gleason problem for the kernel collection {K(I)•...• K(d)}. 3 the operator B solves the Q-structured {K(1), ... ,K(d)}-Gleason problem for S, and 4 the operators C and D are given by 11 (a) d d Pa C:ffi : ~LLlt2(0), D:u~S(O)u. a=1 ' (a) Pc> a=lk=1 Then we leave it to the industrious reader to check that Theorems 4.8, 4.9 and 4.10 all go through with this block-diagonal modification. Specializing this formalism to the polydisk c.ase picks up exactly the results of Section 3. References 1. J Agler, On the representatIOn of certain holomorphc functiona defined on a polydisc, Topics in Operator ThcOIY: Ernst D. Hellinger Memorial Volume, Oper. Theory Adv. Appl., vol. 48, , Barel, 1990, pp. 47 66. 2. J. Agler and J. E. McCarthy, Nevanl,nna PIck Interpolaton on the bid,sk, J. Reine Angew. Math. 606 (1999), 191 204.
- 55. Exploring the Variety of Random Documents with Different Content
- 56. “We have been a happy family during the siege. The time has now come for breaking up. When we were first invested I said to you, ‘Sit tight and shoot straight.’ The garrison has sat tight and shot straight, with the present glorious result. Many nice things have been said about me at home, but it is an easy thing to be the figurehead of a ship. The garrison has been the rigging and sails of the good ship Mafeking, and has brought her safely through her stormy cruise.” He then thanked the ladies, beginning with the matron of the hospital, whose pluck and devotion could not be sufficiently extolled. Turning to the Protectorate Regiment, he said:— “To you I need say nothing. Your roll of dead and wounded tells its own tale.” Shaking hands with Colonel Hore he thanked him for the assistance he had given him, and to the artillery, under Major Panzera and Lieutenant Daniel, he said:— “You were armed with obsolete weapons, but you made up for these by your cool shooting and the way you stuck to your guns.” The colonel afterwards turned to the British South Africa Police:— “I need not repeat to you men the story of the little red fort on the hill, which Cronje could not take.” And to the Cape Police, under Captain Marsh, he addressed himself as follows: — “You have not been given an opportunity of doing anything dramatic, but throughout the siege you have held one of the nastiest places in the town, where the enemy were expected at any moment, and where you were always under fire.” The colonel next made some graceful remarks to the Town Guard. He compared them to a walnut in a shell; saying that people thought that they had but to break the shell to get at the kernel. But the enemy had learnt better. They had got through the husk and found they could get no hold on the kernel. In conclusion, he announced that any civilians who wished to return to their ordinary occupations immediately might do so. Those who had none to return to, whose billets had been lost or businesses ruined, would be permitted in the meantime to draw trench allowances and to remain on duty in the inner defences. Major Goold Adams was then cordially thanked for all the excellent work he had done as Town Commandant, after which the Railway Division (under Captain Moore) and Lieutenant Layton (who had received a commission for his splendid services) were addressed:—
- 57. “I cannot thank you enough for what you have done. You have transformed yourselves from railway-men to soldiers. Your work is not yet done, because it will be your business to reopen communication and get in supplies.” Mafeking Railway Station—The First Train arriving from the North after the Relief. (Photo by D. Taylor, Mafeking.) To the Bechuanaland Rifles Colonel Baden-Powell exclaimed:— “Men, you have turned out trumps. With volunteers one knows that they have been ably drilled, but there is no telling how they will fight. I have been able to use you exactly as Regular troops, and I have been specially pleased with your straight shooting. The other day, when the enemy occupied the Protectorate Fort, they admitted that they were forced to surrender by your straight shooting, under which they did not dare to show a hand above the parapet.” The chief delighted the juvenile Cadet Corps by giving them their meed of praise for their conduct as soldiers, concluding with, “I hope you will continue in the profession, and will do as well in after life.” He then turned to the outsiders, the Northern Relief Force under Colonel Plumer, which had borne the brunt of the seven months’ fighting, and expressed his regret that they had been too weak to relieve the town “off their own bat.” But he eulogised the splendid work done in bad country and climate. The Southern Force under Colonel Mahon were congratulated on having made a march which would live in history. Their chief was complimented on the magnificent body of men he commanded, while the Imperial Light Horse, associated as it was with memories of Ladysmith, Colonel Baden-Powell
- 58. declared he was especially pleased to see, as these would be able, in consequence of their own experience, to sympathise with the people in Mafeking. So the amazing defence of Mafeking was over! For seven months the gallant little town had withstood every ingenious device of the Boers, and in the end it had come off victorious. The first shot was fired on the 16th of October, and from that day the rumble of bombardment had been the accompaniment of almost every hour between the rising and setting of the sun. And now all was serene and still, and only the battered walls of the once neat little hamlet told the terrible, the glorious tale of British doggedness and British pluck. Lord Roberts Lord Kitchener LORD ROBERTS AND HIS ARMY CROSSING THE VAAL RIVER Drawing by R. M. Paxton, from a Sketch by W. B. Wollen, R.I. HOW THE NEWS WAS RECEIVED BY THE BRITISH EMPIRE.
- 59. For some time the ears of London had been pricked up in anxious expectation. Lord Roberts had promised to relieve Mafeking by the 18th of May, and the Field-Marshal was known to be punctuality personified. All the town remained in a state of suppressed excitement, little flags were selling like wildfire, and big flags were being got into readiness for the great, the longed-for word. Early in the morning of the 17th the papers were anxiously perused, and man asked man if any news had leaked out. The 18th arrived. Nothing was known. The War Office maintained its adamantine calm. The day grew middle-aged, almost old—then, as the shutters were about to go up (twenty minutes past nine was the exact hour), one telegram of Reuter’s fired the fuse, and London, followed presently by the whole British Empire, was ablaze with excitement. The flame, like most flames, broke out almost unnoticed. Some one on a cycle —some one in a cab, heard the glorious three words, and sped breathless to carry the contagion of his rapture far and wide. Street after street began to smoulder—to glow; and, presto! the town was one vast conflagration! Such a furnace of patriotism had never been seen within the confines of the staid metropolis. By ten o’clock the populace of one consent had run wild into the streets—the houses were too cramped to hold them—they ran wild, roaring and yelling and shouting and singing, passing into the heart of the Capital in dense armies—passing? nay!—for soon none could pass, but had merely to be propelled good-humouredly by the compact mass that surged apparently to no destination whatever. Whence came the clamouring hosts it was impossible to say—they seemed to rise from the earth, so rapidly, so mysteriously, did their numbers increase. Liberty, equality, fraternity, was the motto of this memorable night. All ages, and ranks, and sexes were linked together in the bonds of sympathetic patriotism—countess or coster, duke or drayman, it was all one—an identical beam of triumph imparted a relationship to every British face. Minutes had scarcely grown into hours before the Union Jack fluttered from every window, from every cart and ’bus, from every hand, and the roar of human joy was as the roar of the ocean in a tempest. At the theatres, as at the railway stations, the crowds heard and wondered only for a moment, for the electrical news got into their midst, and they on the instant took up the cry and the cheer, and repeated them with all their might. Indeed, theatrical performances were suspended while the joyous audiences sang and re-sang “Rule, Britannia” and “God Save the Queen,” and then, unsatisfied, tore into the open to let off steam as it were, and view a sight which never before has been witnessed, and probably never again will be visible in the precincts of London Town. The Mansion House, where the display of the message had caused a huge concourse to assemble, was next besieged, and the old walls literally shook with the mighty roar of the multitude. The “National Anthem” swelled out thunderously with volume that was almost awe-striking as the
- 60. combined voice of a Handel Festival, and shouts for the Lord Mayor grew and grew, and became deafening as that honoured citizen and splendid patriot showed himself. He then delivered the following speech: “I wish the music of your cheers could reach Mafeking. For seven long weary months a handful of men has been besieged by a horde. We never doubted what the end would be. British pluck and valour when used in a right cause must triumph. The heart of every one of you vibrates with intense loyalty and enthusiasm, I know, and the conscience of every one of you assures you that we have fought in a righteous and just cause.” The crowd, incapable of silence for very long, broke into “Rule, Britannia,” and when this outburst of emotion was expended, the Lord Mayor continued: “We have fought for our most glorious traditions of equality and freedom, not for ourselves alone, but for the men of all those nations who have settled in South Africa and who were under the protection of the British flag.” Three cheers for Colonel Baden-Powell were then called for, and three for Lord Roberts, and these having been heartily given, he said: “The people of Bloemfontein and Mafeking are now singing ‘God Save the Queen’; you can do it for yourselves.” This they proceeded to do not once but twenty times through the livelong hours of the night. Meanwhile the following practical telegram was despatched by the Lord Mayor:— “To Baden-Powell, Mafeking, via Cape Town. “Citizens London relieved and rejoiced by good news just received. Your gallant defence will long live in British annals. Cable me what money wanted for needs of garrison and inhabitants after long privations. “Alfred Newton, Lord Mayor.” At the same time a huge portrait of Colonel Baden-Powell was displayed in front of the Mansion House, and the strains of “God Save the Queen” and “Rule, Britannia” were now intermingled with the lively tune of “For he’s a jolly good fellow.” These combined choruses were echoed and re-echoed, and carried along like a gigantic stream of sound into the suburbs of London, into sleeping Kensington and remote Clapham, so that men and women turned in their beds—sat up, terrified at first, then realising the situation, gave up thought of rest, and listened with swelling hearts to the triumphant din. And so, on and on—through the night till morning broke! Then, the whole face of London seemed transmogrified. National emblems— red, white, blue, yellow, green, stars and stripes—draping the houses and festooning the roads, gave the town the aspect of one huge bazaar. Balconies were decorated, awnings thrown out, and in some cases, to give a touch of
- 61. realism, bathing towels[6] were hung from the verandahs. People passing by, and ignorant of the double meaning of the curious drapery, shrugged their shoulders, scoffed—then, awakened by a flash of illumination, looked again and broke into renewed cheers. Before the dwelling of the mother of the defender of Mafeking a vast crowd collected, wielding flags and laurels, and displaying in their midst the bust of the hero with a British lion crouching at his feet. Cheers rent the air, and increased in volume when the proud parent of this splendid Briton appeared on the balcony and acknowledged the demonstration. The glad tumult in front of this point of attraction continued throughout the day, people coming from far and wide here to vent their ecstasy of enthusiasm—some in shouts, many in tears. By nightfall, the whole Empire was pouring forth its excitement in congratulatory telegrams, for, four minutes after the receipt of the intelligence in London the news had passed over the Atlantic cables and was in the New York office of the Associated Press, whence it was forwarded to the farthest limit of the North American Continent. Canada, New South Wales, Sydney, and all the other colonies whose bravest and best had contributed to the great doings in the Transvaal, were now aglow with bunting and illuminations. Church bells pealed, processions passed shouting and rejoicing, ships were dressed from truck to taffrail, and prayers and anthems of praise were got ready to be offered up on the following day at all churches. Thus, for a brief space, was seen a vast concourse of millions of souls of differing opinions, customs, and creeds, diffused even to the remotest corners of the British-speaking world, yet closely united by a bond of fraternal sympathy in consequence of the triumph of British manhood in the most unique ordeal that the loyalty of any nation has been called upon to endure. FOOTNOTES: [5] See Vol. III. p. 39. [6] The hero of Mafeking at Charterhouse was nicknamed “Bathing Towel.”
- 62. CHAPTER VI FROM KROONSTAD TO JOHANNESBURG From the 12th to the 22nd of May was spent by the main army, at Kroonstad, where, owing to sickness and other causes, a halt was obligatory. It was necessary that supplies should be collected, an advanced depôt formed, the railway repaired, and the safety of both flanks secured. Meanwhile, efforts were made to protect the farmers who had surrendered from the revengeful tactics of the Boers. Lord Lovat’s gillies arrived at Kroonstad and met with the approval of the Commander-in-Chief. General Hutton, with a force of mounted infantry, had reported an attack on Bothaville and the capture of three commandants and about a score of Zarps, from their hiding-place near Smaldeel. On the 20th, the 1st Cavalry Brigade marched out from their camp near Kroonstad, to open up the country on the left of Lord Roberts’s main advance along the western fringe of the railway. They were accompanied by the 4th Cavalry Brigade (7th Dragoon Guards and 8th and 14th Hussars), and supported by General Hutton’s Brigade of Mounted Infantry (Canadians, Australians, and New Zealanders). On the 21st, the cavalry seized the drift at the confluence of the Honing Spruit and the Rhenoster; and on the 22nd, Lord Roberts and the main army, leaving only the 1st Suffolks behind, marched from Kroonstad to Honing Spruit, the third station to the north, and some eighteen or twenty miles off. General Ian Hamilton, after a series of engagements with De Wet’s hordes, from Lindley, onwards, had secured an advanced position at Heilbron, while the cavalry division had moved up, crossed the Rhenoster River, and threatening the right rear of the enemy had forced the Dutchmen to leave a strongly-entrenched position on the north bank of the river. The presence of French and Hamilton to west and east of them had served to unnerve the hostile hordes, who now had our cavalry within twenty miles of either flank. They spent their bellicose ardour by destroying some miles of railway, the bridge over the Rhenoster, and some
- 63. culverts, and then flying in hot haste before the vast machinery of the advancing army, to a new point of defence some twenty miles in front, a point which promised shortly to become equally untenable. THE GREAT ADVANCE: ROYAL HORSE ARTILLERY (CAVALRY DIVISION) CROSSING THE VAAL Drawing by R. Caton Woodville The following casualties took place in the Winburg Column, May 21st:—New South Wales Mounted Infantry—Wounded severely, Lieutenant A. J. M. Onslow, 1st Royal Irish—Lieutenant M. H. E. Welch. On the 23rd, Lord Roberts and his majestical and magnificent apparatus of war, its thousands of gallant souls, its multiplicity of vehicles, its endless supplies and zoological train, encamped on the south bank of the Rhenoster River. The Boers, apparently demoralised in their preparations for resistance, and having had their left flank turned by Hamilton at Heilbron, were now continuously “on the run.” Meanwhile burghers hourly came in to surrender arms and ammunition, the last vestige of truculence having evaporated. The Boer Government telegraphed to Lord Roberts offering to exchange an equal number of prisoners on parole, and threatening if the offer should be refused to remove from Pretoria to some other district the 4000 prisoners now confined there. As to the fate of the Johannesburg mines there was
- 64. considerable uncertainty; reports declared they would be destroyed in the event of entry to the Transvaal by the British, and also that the town itself would be defended, as defence works were being rapidly pushed forward, guns got into position, and trenches and defences constructed. On the other hand it was stated that, on hearing of the threat to destroy the mines and possibly the town, Commandant Louis Botha had hastened to the President, and in a stormy interview had asserted his intention, if such a thing were contemplated, himself to defend Johannesburg from such an act of vandalism. He concluded by denouncing the diabolical intention and saying, “We are not barbarians.” Mr. Kruger did not argue the subject—possibly his conscience tweaked him on the subject of barbarity—but gave in. Terrible altercations were daily taking place between the Boers, the Free Staters, and their mercenaries, and the burghers were inclined to throw all the blame of defeat on the Hollanders who had brought about the war and left the Boers to bear the brunt of the loss to life and property that hostilities entailed. These were merely reports, but they served, as the passage to the north proceeded, to show which way the wind blew. On the Queen’s birthday the 4th Brigade of cavalry crossed the Vaal near Pary’s Drift, and the 1st Brigade at a drift farther east of Pary’s, while General Ian Hamilton’s column was ordered to move towards Boschbank still higher up. They arrived just in time to save the coal-mines from being destroyed. The operation of crossing the Vaal was one of the most risky that has been undertaken in the campaign, as the road down to the drifts led through about six miles of mountainous country forming a narrow pass, well suited to Boer tactics. Fortunately, although the Boers were seen hovering in the vicinity, the arrival of the cavalry was unexpected, and they made no effective resistance. It will be seen that here the distribution of the advance underwent a change. General French adhered to his original course on the left, but General Hamilton, screened by Gordon’s Cavalry, crossed in front of the main army, and concentrated near Vredefort on the west, thus preparing a little surprise for the Boers, who were collected in their thousands opposite Engelbrecht Drift in the expectation that the British General would continue to proceed towards the north. Meanwhile, the cavalry, to a desultory accompaniment of musketry, was engaged in securing the approaches to Lindique Drift, over which the baggage had to pass. On the 26th, Colonel Henry’s Mounted Infantry, and the Bedfordshires, crossed at Viljoen’s Drift and there encountered an Irish- American rabble in act of injuring the coal-mines and bridge; and the wreckers —an alcoholically-valiant gang of hirelings—speedily made off, leaving behind them three days’ supplies, which came in most handy for the benefit of the
- 65. troops. By this time General Hamilton had reached Boschbank, and Lord Roberts had arrived at Wolve Hoek. The Cavalry Division, finding the force of Mounted Infantry had moved to Vereeniging—and thus opened up communication with Lord Roberts’s main advance—flew on. On the evening of the 27th they seized the head of the horse-shoe of hills wherein the Boers in large numbers had ensconced themselves. This dashing exploit was attended with the loss of only one Scots Grey and one Carabineer wounded. The position thus gained overlooked the Boers’ main position at Klips Wersberg, defending Johannesburg. While this was going on (on the 27th) Lord Roberts, with the 7th and 11th Divisions, crossed the Vaal facing Vereeniging, and encamped on the north bank, and found vacated several intricately prepared positions whence the Boers had intended to offer opposition. They had abandoned position after position at the approach of one or other of the great feelers of the big British machine that threatened to surround them. The fact was, this enormous army was moving as an avalanche—stupendous and strong—an avalanche that swept all things before it. Horses and men were in splendid fettle, their spirits were rising, their confidence intense, and all endeavoured to emulate the example in activity set them by the Field- Marshal, who, like a young man of thirty, was up before dawn and working hard till sundown. In spite of the cold nights—especially trying after the heat of midday—the Commander-in-Chief looked healthy and well, while his troops, who had marched magnificently in trying circumstances, needed no finer eulogy than to be described as worthy of him. A grand march of twenty miles brought the main army on the 28th, to Klip River, within eighteen miles of Johannesburg—a march so rapid and so well organised that the Boers, who had prepared a delicate salute of five guns with which to welcome the troops, had barely time to hustle their weapons into the train and steam off as some of the West Australian Mounted Infantry dashed into the station! These smart Colonials were very much to the fore all day and showed a vast amount of dash and dexterity. Major Pilkington and a patrol of some thirty of them were moving in advance of the 11th Division in hope to find a suitable drift for the passage of troops and guns across the Klip River. The drift was discovered, but also the Boers—a posse of them hovering among the kopjes that flanked the road. Without ado, the little party prepared themselves for the worst, spreading themselves, rifles in hand, to protect the position they had gained, a position of some importance, since it commanded bridges about a mile and a half to east and west of the road. The party
- 66. divided into two groups, arranged themselves at each bridge, and endeavoured to make a line—a very thin line—as a uniting link between the groups. It was somewhat like the fable of the frog that tried to blow himself out to the size of a bull—but in this case the minute object’s pretence was successful; the thirty isolated men deluded the Boers, and caused them to believe that these sturdy defenders of the drifts were supported by a huge force in reserve. Blazing away with their rifles, the Dutchmen attacked the small party, and an uneven contest commenced and proceeded till dusk. Lieutenant Porter, while directing some operations, was wounded, but fortunately at this juncture there came to his rescue some guardsmen, who were escorting a convoy, and these, owing to the gallant manner in which the drifts had been held, managed in the darkness to get their convoy into safety, and enable the Westralians, whose work was accomplished, to “silently steal away.” Meanwhile, during the whole day, some ten miles to the left—on the west of the railway—sounds of animated knocking portended much activity on the part of Generals French and Hamilton in the neighbourhood of Syferfontein and Klip River. General French was engaged in a reconnaissance in force of the enemy’s position. After drawing the fire of all the Dutch guns, and consuming a good deal of powder, the casualties on the part of the cavalry were small— about five—mostly Inniskillings. On the 29th of May, part of the Cavalry Division, General Ian Hamilton’s Mounted Infantry, the 19th and 21st Brigades, and some Colonials who had moved parallel to the main advance since it left the Vaal, found themselves about twelve miles south of Johannesburg. East of Doornkop some 4000 Boers, with six guns, had taken up a menacing position, strengthened with various natural obstacles, while the ground had been blackened with grass fires to afford an effective background to approaching kharki. The troops, supported by the guns, at once steadily advanced to attack the Boer centre, while Generals French and Hutton operated on the west to turn the right flank of the position. After an hour’s smart fighting the infantry were able to push on, Porter’s brigade having ridden five miles to the west, and turned the enemy’s right, while the infantry, with fixed bayonets, had driven the enemy from every cherished kopje. In the attack, the Gordons in the centre of the right, the City Imperial Volunteers in the centre on the left, advanced gradually on the Boer position. The gallant nature of the advance over the burnt and blackened ground, which made the infantry into targets for the foe, excited the admiration of all. Grandly the Gordons flung themselves upon the enemy, in spite of the Boer guns and “pom-pom,” that dealt death and destruction among their numbers. Seventy of the dashing fellows dropped, and the only consolation for so great a loss was, that by nightfall 6000 Dutchmen were
- 67. scudding away in the darkness, while General Hamilton was bivouacking on the ground seized from them, and Generals French and Hutton, who had turned the right flank of the position, were threatening Krugersdorp. The conduct of the City Imperial Volunteers was magnificent, and to them, as well as to the Gordons, much of the credit of the day’s work was due. They behaved as skilled troops, taking cover with great ingenuity, and returning the attacks of the enemy with amazing coolness and precision. Their sustained volleys succeeded in clearing out the Boers immediately in front of Roodepoorte. Commandant Botha—not Louis Botha, but a kinsman—with a hundred foreign and Irish subsidised sympathisers, was captured, and, in addition to these, a Creusot gun and twelve waggons of stores and ammunition were secured. The losses among officers in this engagement were comparatively few. Captain St. J. Meyrick, 1st Gordon Highlanders, was killed. Among the wounded were:— City Imperial Volunteers—Capt. G. W. Barkley. 1st Gordon Highlanders—Capt. G. E. E. G. Cameron, Lieut.-Col. H. H. Burney, Capt. P. S. Allen, second Lieut. A. Cameron, Surg.- Lieut. A. H. Benson, Dr. R. Hunter. Vol. Co. Gordon Highlanders—Capt. J. B. Buchanan, Lieut. J. Mackinnon, Lieut. H. Forbes. Royal Army Medical Corps—Lieut. A. H. Benson. 2nd Duke of Cornwall’s Light Infantry—Lieut. H. W. Fife (since dead). 10th Hussars—Lieut. T. Lister. During General French’s operations near Klip River, on the 27th, 28th, and 29th, the wounded officers were:— New Zealand Rifles—Captain Palmer. 7th Dragoon Guards—Major W. J. Mackeson, second Lieut. G. Dunne. Capt. D. L. MacEwen, Cameron Highlanders, attached to Intelligence Department, was taken prisoner.
- 68. GENERAL IAN HAMILTON THANKING THE GORDONS FOR THEIR ATTACK AT THE BATTLE OF DOORNKOP Drawing by S. Begg To return to the main advance on this day (29th). While Generals French and Hamilton were engaging Botha and his hordes outside Johannesburg, turning their flank wherever they posted themselves, Lord Roberts decided to pursue boldly the programme of his main advance upon the enemy’s East Rand and Pretoria communications, a programme which was as faultlessly and rapidly carried out as it was skilfully conceived. From the neighbourhood of the Klip River the troops pushed on rapidly to Germiston without meeting with serious opposition. So swiftly were the movements executed that the nimble Boers were beaten at their own game, and had to turn tail without removing the whole of the rolling-stock. Thus, the Commander-in-Chief came at once into possession of the Junction connecting Johannesburg with Natal, Pretoria, and Klerksdorp by railway, and through a piece of splendid strategy Boer resistance was paralysed, and the railway system of the State was brought completely under his control. Any concentration of forces in Pretoria or on the fringes was now practically impossible. The history of the hurried capture of this vital strategical position was inspiriting. Colonel Henry, with the 8th Mounted Infantry, started at dawn with orders to seize Elandsfontein at all costs. The 3rd Cavalry Brigade in support made a detour to the east towards Boksburg, in a
- 69. direct line to Pretoria, followed rapidly along the line by Pole-Carew’s and Tucker’s Divisions. The object of the somewhat wide easterly move was to outflank the enemy’s defensible positions and secure the communications to Pretoria, and thus cut off and isolate the force prepared to check the advance of the British. Just as the advance guard neared the Natal line, a train was seen conveying half of the Heidelberg Commando from Volksrust to the north. It was impossible to arrest it, but after firing on the departing machine, the troops proceeded to demolish the line and secure the Natal communications. The Mounted Infantry which, owing to the uselessness of the Klip River Bridge, were without artillery, were now assailed by a party of Boers with guns, who had ensconced themselves in the ridges which menaced the southern road, but nevertheless they pressed forward bent on obeying orders and gaining Elandsfontein. They pushed ever on and on till the great city, the monstrous hive of gold- getters, the scene of Boer despotism and Uitlander servility, became visible from the rolling hills. Momentarily they expected to hear a roar, to see a flare and an upheaval, and to know the worst had come—the mines had been destroyed! But all was silence. The huge town, surrounded in places by a blanket of smoke, seemed slumbering on the bosom of the undulating downs. In the distance, however, the station showed active. Trains were steaming off to Pretoria. Others with their steam up were preparing to follow. These trains must be arrested, and their freight captured. It was a case, unfortunately, of horse-flesh versus steam. But still it was worth the venture! Off went a section of the Yorkshire Mounted Infantry, galloping like fury to the station, while the main body made for Boksburg; and the Australians, toolless, tore to Knight’s Station, and there piling up trollies, boulders—anything, in fact, that came to hand—blocked the line. They were pelted by hidden Boers, but fled carefully to cover after accomplishing their object. Meanwhile, some of the Yorkshire Mounted Infantry had seized the station, and, with it, three locomotives whose steam was up ready for departure. But the enemy were in strength there —they were at least strong in proportion to the twenty dashing Yorkshire men who had plunged into the mêlée, and these gallant fellows found themselves in a critical position, fighting like demons for their hardly-earned prize with desperate men, whose sole source of salvation lay in the locomotives that stolidly panted and wheezed in utter disregard of the fierce fight raging for their possession. Then, with almost theatrical precision, a vast procession was seen to be approaching: a river of kharki flowing down the southern slopes into the Rand. It was the Mounted Infantry from Boksburg and the Infantry Division—the goodly Grenadiers leading—pouring in their numbers to the rescue of the gallant little band! Thus by nightfall one of the most fateful of the operations of the war was concluded, and Johannesburg was virtually seized without the wrecking of a mine and with little loss of life. During the operations Captain MacEwan, Cameron Highlanders, and Lord Cecil Manners (correspondent to the Morning Post) were taken prisoners. Lieutenants Pepper, West Australian Mounted Infantry, Beddington, Imperial Yeomanry, and Forrest, 1st Oxford Light Infantry, were wounded. Immense crowds, surprised to find that the struggle was a matter of hours and not of days, watched the fighting from west and east corners of the town, and the shock of the fall of Elandsfontein disorganised their plans and demoralised themselves. While this was going on, the Cavalry Division had advanced through the gold mines, having Johannesburg on their right, and was encamped on the west of the town, keeping a wary eye on the Boers, who were fleeing hot-foot to Pretoria. Within the City of Gold, all was turmoil. On the discovery of the situation there followed a violent up-rising. The Kaffirs, on seeing the Boers repulsed, rushed to the Jews’ houses to loot them, and the foreign contingents immediately set out on a species of internal invasion,
- 70. breaking open shops and stores and houses, and throwing out of doors and windows goods collected for the benefit of needy burgher families. The uproar, however, was speedily suppressed by the firm measures of Dr. Krause. In answer to the flag of truce sent in by the Field-Marshal, this official went out to meet him. There being still many armed burghers in the place, the Transvaal Commandant requested Lord Roberts to postpone his entry for six hours. To avert disturbance this arrangement was agreed to, and Lord Roberts decided to postpone till the 31st his entry into the conquered town. So Johannesburg was ours! The advance, which appeared to be so rapid, straightforward, and simple, owed these qualities to Lord Roberts’s splendid, almost prophetic, instinct for gauging the enemy’s expectations with a view to disappointing them; to his strategic manipulation of his cavalry and mounted infantry, and to the magnificent marching capability of the infantry. Everywhere, the Boers had fenced themselves across the route, sometimes extending their line of defence for twenty miles or more, and everywhere, in dread of having one flank or the other turned, they had been kept oscillating between stubborn resistance and rapid flight till their nerves had given way, and they had scuttled back and back to their undoing. At the Vet, the Zand, the Valsch, the Rhenoster, and the Klip Rivers, they had cunningly prepared themselves, till, with the infantry menacing them in front and the cavalry and infantry threatening both flanks, they had realised that retreat was inevitable. Their last hope had been set on the city of mines; and now from thence, a routed, raging rabble, they were fleeing in despair. The splendid progress of the infantry was a remarkable achievement, of which enough cannot be said. It was no mere feat of pedestrianism. It was a march in face of an enterprising enemy, and harassed with discomforts sufficiently multifarious to try the endurance of a Socrates. A scorching sun by day and a frigid temperature by night, occasional sand blasts rendering drier than ever parched throats already dry as husk from the tramp through a sand- clogged and almost waterless country, were but items in the programme. If water there chanced to be, it was ochreous and fouled by the passage of many quadrupeds, and such food as there was—bully beef and adamantine biscuit—demanded the jaws and digestion of an alligator. Yet these sturdy fellows plodded along, lumbering through sand drifts and squelching in mire and morass, or laid themselves to rest on the hard or soggy ground with a philosophy so devil-may-care as almost to fringe on the sublime. With unquenchable gaiety, they had accomplished a march of 254 miles (the distance from Bloemfontein to Elandsfontein) in eighteen days, giving as an average fourteen miles a day. (This calculation naturally excludes the ten days’ halt at Kroonstad.) From Kroonstad to Elandsfontein, a distance of some 126 miles—covered in seven days (22nd to 29th)—marching had gone forward at the rate of eighteen miles a day. Napoleon’s much vaunted march from the Channel to the Rhine in 1805 showed an average of sixteen miles a day, when the distance traversed was 400 miles, and the time taken twenty-five days. But that march, unopposed throughout, was comparatively plain sailing. Quicker forced marches have been known,[7] but in the present case the march was continuous, and may be said to beat all records of rapid marching under equally inconvenient conditions. The twenty-four hours were allowed to pass. Then, at the entrance of the town Dr. Krause met the Commander-in-Chief, and rode with him to the government offices, and introduced to him the heads of the various departments, all of whom were requested to continue their respective duties till they should be relieved of them.
- 71. To those who had never seen Johannesburg the first glimpse was a surprise. Strangely incongruous did it seem to move from the isolation and rugged simplicity of the open veldt to the centre of a large and peculiarly civilised town. The note of modernity was sounded on every side. Buildings more than magnificent greeted the eye accustomed only to homely farms and mushroom staadts. Tramways ribbed the streets, electric lights gleamed a whiter glare than moonbeams, and nineteenth-century luxury, and in some cases refinement, were in evidence at every turn. But the public buildings were closed, and the handsome shops boarded up for precaution’s sake, while the streets were thinly populated, owing to the fact that many of the British sympathisers had been expelled, and the Boer community was on commando. THE CITY OF LONDON IMPERIAL VOLUNTEERS SUPPORTING GENERAL HAMILTON’S LEFT FLANK IN THE ACTION AT DOORNKOP ON THE 29TH OF MAY Drawing by C. E. Fripp, R.W.S., War Artist But though at first the place was deserted, by degrees people began to trickle in, and by the time the square in front of the government buildings was reached there was a goodly throng. The Vierkleur was still flying when Lord Roberts, at the head of General Pole-Carew’s division, marched into the town; but presently the keys were formally surrendered, the flag was hauled down, and a small Union Jack, worked by Lady Roberts, was hoisted in its place. At the conclusion of the ceremony the rousing strains of the Guards’ band were heard, and the 11th and 7th Divisions marched past, with the Naval Brigade, the heavy artillery, and two Brigade Divisions of Royal Horse Artillery. General Ian Hamilton’s column and the Cavalry Division and Mounted Infantry were too far away to take part in the proceedings.
- 72. Vaal River Deviation Bridge at Vereeniging, nearly completed. (Photo by W. H. Gill, London.) It was an impressive spectacle; one ever to be remembered. From afternoon till night, troops —great, brawny, bronzed, and workmanlike Britons—came clanking in procession through the town, while from balconies and windows banners and flags were waved, and gay ladies, many of them Englishwomen, wild with excitement and enthusiasm, threw down flowers and sweets and cigarettes to give vent to their unrestrained joy. Far into the evening the stream of kharki continued ceaselessly to flow under the magnesian rays of the electric lights till the infantry had passed to their camp, three miles to the north, and Lord Roberts had settled himself at Orange Grove. FOOTNOTES: [7] See vol. iv. p. 41.
- 73. CHAPTER VII GENERAL RUNDLE’S MARCH TO SENEKAL While Lord Roberts was moving from Bloemfontein, co-operative action was being taken elsewhere. On the 2nd of May the Boers evacuated Thabanchu and trekked towards the north, and on the following day General French, leaving General Rundle in command, started to join Lord Roberts’s main scheme. Soon after General Brabant joined General Rundle’s force. On the 4th, General Rundle moved forward from Thabanchu, attacked the enemy, captured their positions, and headed them eastward. There was little hard fighting, the General’s movements being mostly carried out with so much celerity, and strategical and tactical skill, that the enemy, seeing British forces apparently in strength everywhere, judged it advisable to move from post to post rather than run the risk of being mopped up. On Friday, the 11th of May, Colonel Grenfell, with the 2nd Battalion of Brabant’s Horse, attacked the Boers at Ropin’s Kop, but was overpowered by the enemy and forced to retire, with several wounded. On the following day, Saturday, he, however, drove the Boers out of their position, and captured Newberry Mills at Leeuw River, thus depriving the Dutchmen of an immense store of flour and grain which it had been their ambition to seize. This smart piece of work was accomplished almost without casualties. While these operations had been going forward, some 500 of the Yeomanry had occupied the northern slopes of Thaba Patacka, a position whence they hoped to attack the Boers who might be slinking off in the direction of Basutoland. General Boyes, on the west, was equally active, to the dismay of the Boers, who, owing to General Rundle’s clever strategy, imagined the British held a front of over twenty miles. On the 13th of May General Rundle advanced to Brand’s Drift, twenty miles to the north-east, taking prisoners and accepting the surrender of many Free-staters, who were perished with cold and exposure, and sickened by defeat. Meanwhile, General Brabant, performing like operations, was slowly moving northwards. On the night of the 15th, Ladybrand was occupied by a force of the Glamorganshire Yeomanry, and thus the two Generals maintained possession, by magnificent strategic moves, of the whole southern corner, which is practically the granary of the Free State, gradually scaring away the enemy from the country through which they passed. On the 24th, a simultaneous movement was made, Brabant’s Colonials marching to occupy Ficksburg, while General Rundle with General Campbell’s Brigade, followed by that of General Boyes, proceeded towards Senekal. During the march an unfortunate incident took place. On reaching Mequaling’s Nek, a rumour reached General Rundle that the Boers were in retreat from Senekal, consequently on the next day, the 25th, Major Dalbiac and Major Ashton, R.M.A. (Intelligence Officer to the
- 74. Division), were ordered to investigate the nature of the water supply, and to find a camping ground in the neighbourhood of the town. Major Dalbiac and a company, mainly composed of Middlesex Yeomanry, accompanied Major Ashton as escort, and the party left at dawn and proceeded to Senekal. Here they encountered apparently peaceful inhabitants, and were entirely ignorant of the fact that the Boers had merely vacated the place for the purpose of hiding themselves in a hilly coign of vantage, which practically commanded the streets of the town. Major Ashton proceeded with the inquiries he was deputed to make, and received from a citizen the keys of the official buildings, which had been left by the Landdrost, who with the postmaster and other responsible persons had decamped. Then came the surrendering of arms, and while this was going on, suddenly, without warning, a heavy fusillade was launched at the Yeomanry who formed a group round Major Ashton. For a moment chaos reigned; then all sprung to action. The Boers, delighted at their surprise, blazed away fast and furious, while the two Majors, gathering together their little band, made hurried arrangements. Major Ashton, with some ten men, enclosed himself and promptly commenced firing on the incoming enemy, while Major Dalbiac with a score of the Yeomanry, dashingly galloped off in hope of taking the enemy in rear. But the Boers were many and the unfortunate Yeomanry quite outnumbered. No sooner had they wheeled round the hill, than rifles poured a withering fire on them. Six horses dropped even as the men dismounted, and the ground, open and quite devoid of cover, was strewn in one moment with the slain and the suffering. Major Dalbiac almost instantaneously dropped dead. He was shot through the neck, and four men shared his fate. Lieutenant Hegan Kennard, wounded in the face, was in a desperate plight, while nearly all who remained were injured. Some half-a-dozen men had been sent back with the horses on the first outbreak of the attack, and these only of the valorous band escaped. Meanwhile news of the ambuscade had been carried to General Rundle, who instantly ordered off the Wilts Yeomanry, 2nd Grenadiers, and 2nd East Yorks, with artillery, to the succour of the unfortunate party. These arrived in time to save Major Ashton. He had fortunately occupied the side of the town towards which the British approached, and the Boers, at the first sound of the guns which had been directed against the kopje where they had ensconced themselves, made off with all possible speed. By the time General Rundle had neared the town, it had resumed its pristine state of innocence, and the inhabitants were preparing effective demonstrations of loyalty. In the evening the remains of the unfortunate dead at the foot of the hill were recovered, and it was found that Major Dalbiac’s body had been rifled by his dastardly opponents of every article of value, and even the ribbons of his medals were missing. On the 26th, General Rundle with the 8th Division entered the town and formally took possession of it. The remains of Major Dalbiac and the four men of the Middlesex Yeomanry who were killed in the unfortunate affair were buried with military honours, the General and Staff attending the funeral. A patrol of the Hants Yeomanry, while out scouting, got in touch with the enemy, and escaped by what is called the skin of their teeth. Many had very narrow escapes, and one man was killed. Sergeant-Major Foulkes, whose horse was shot under him, was saved through the gallantry of Private Andrews, who returned and bore off his dismounted comrade, while Captain Seely and others behaved in like manner to ensure the safety of those left without mounts. GENERAL COLVILE AND THE HIGHLAND BRIGADE
- 75. Of the Highland Brigade since the tragedy of Majersfontein and the smart fight at Koodoesberg little has been said. Their brilliant march and action before Paardeberg, in which General MacDonald was wounded, served to demonstrate the stuff of which they were made and to restore their self-confidence and zest for battle. Lord Roberts’s gracious speech, delivered at the camp, recalling his pleasant association with the Brigade in India, where “they had helped to make him,” and saying that as he had never campaigned without Highlanders, he “would not like to be without them now,” had done much to heal the sore which still rankled in many breasts. HAULING DOWN THE TRANSVAAL FLAG AT JOHANNESBURG Photo by Lionel James On the 1st of May the 9th Division marched from Waterval, picked up the Seaforths at the waterworks, and also the Highland Light Infantry from Bloemfontein. The Division, of which the Highland Brigade, the Seaforths, Black Watch, Argyll and Sutherland Highlanders, and Highland Light Infantry formed the infantry battalions, with the 5th Battery Royal Field Artillery, two naval guns (4.7 calibre), and a company of Engineers, was under the command of Major-General Sir H. Colvile. The Highland Brigade was commanded by General MacDonald. The Eastern Province Horse, a smart and sportsmanlike set of mounted men, numbering about a hundred, also accompanied the force, and did valuable service in scouting. Later on the force was joined by Lovat’s Scouts, but not till the advance was well under way. On the 4th the Brigade bivouacked at Susanna Fountain after an animated tussle with the enemy, who were finally routed by the gallantry of the Black Watch. The Division reached Winburg, as we know, on the 6th, and remained in possession till the 17th. Then, the Black Watch and the Argyll and Sutherland Highlanders advanced, leaving behind them the Highland Light Infantry and Seaforths in the town. On the following day the Zand River was crossed. Ventersburg was entered without opposition, the way having been previously swept by Lord Roberts’s force which had arrived there on the 10th. Here there was a brief halt—a much needed one—as the troops had marched thirty-four miles in 18½ hours.
- 76. On the 23rd they proceeded towards Lindley, and were joined en route by the remainder of the divisional and brigade troops. On the 24th the troops reached a point east of Bloemspruit, where they bivouacked, and the next day brought them into the teeth of the enemy, who were hiding in a ridge at Maquanstadt. From this point the Dutchmen were driven by the Seaforths, who from thence proceeded to a peaked kopje which commanded the water supply, a position which was at once vigorously contested by the Boers. After a hard fight, in which one officer and three men were wounded, the Seaforths succeeded in occupying the position. Here they were joined by the Black Watch and the 5th Battery of the Royal Field Artillery, the rest of the troops remaining behind at Hopefield till the 26th. At Bloemberg, a horseshoe-shaped ridge near Koorspruit (an affluent of the Valsche), the Boers were found strongly posted, and no sooner had the Black Watch appeared than they were greeted by a crackling cross-fire that sent them quickly to cover. Here they held the enemy while a wide turning movement was made to the right. The inner side of the horseshoe position was attacked by the Seaforths, while the outer was assailed by the Argyll and Sutherland Highlanders under Major Urmston, who deftly approached the stony eminence which concealed some sixty of the enemy, and charged with such force and impetuosity that presently the entire position was vacated, and the whole body of Boers, some 1000 in number, were seen racing over the boulders with more than their usual agility. The Bloemberg Ridge gained, it was promptly occupied by Black Watch and Seaforths. By midday the passage of the hill was accomplished, and by 4 p.m. the troops had reached Lindley. The expedition had cost them two killed and eleven wounded. The Highland Brigade crossed the Valsche River and bivouacked north of the drift on the Heilbron Road. Still more north—about two miles—went two companies of Argyll and Sutherland Highlanders to ensconce themselves on a kopje which commanded the road towards Heilbron. On the afternoon of the 27th the advance was continued. The Highlanders crossed the Rhenoster River at Mildraai, and on the following day, 28th, moved still further forward till stopped by the presence of the enemy, who barred the line of march on the north of Roodeport. The Highland Light Infantry—the advanced guard—were deployed and sent to seize some kraals about 1200 yards from the enemy’s position, which sprayed itself over about six miles of country. One company was detached to hold a hill on the right front, supported by the Black Watch, while the Seaforths attempted a turning movement to the left and the Argyll and Sutherland Highlanders guarded the rear and both rear flanks from a point of vantage on Spitzkop. The artillery blazed copiously for an hour, while the Boers also made animated resistance, but after good sixty minutes of assault the enemy gave way, and the Seaforths succeeded in getting round the right flank, while the Highland Light Infantry and Black Watch gained the centre of the now deserted ridge. But the Boers had only scuttled to other ridges whence they could let loose Pandemonium with increased vigour. Thus the Highlanders came in for murderous attention in front, rear, and flank. Presently to their rescue went the invaluable naval guns, snorting vengeance, and determining to show that, though the Field Artillery became outranged and impotent, there was laudable lyddite to save the situation. On this, and with startling velocity, the Federals removed themselves, and they were stimulated in their departure by long-range volleys from the Highland Light Infantry. While the Dutchmen were speeding into the unknown, the Highlanders triumphant were advancing to a position north of Marksfontein. Having crossed the drift they bivouacked on the other side, while the ox transport moved up to the shelter of their wing. The day’s work was not without its pathetic side, for thirty men and three officers were wounded, while two gallant
- 77. Highlanders were among the slain. The wounded officers were: Seaforth Highlanders—Lieut.- Col. Hughes-Hallet, Lieut. Ratclyffe, and Lieut. Doig. At this time the Duke of Cambridge’s Yeomanry were to have met Sir H. Colvile, but owing to their failing by an hour or so to join him on his march up from Lindley they were surrounded, and on the 31st were captured by the enemy. The tale of the disaster is told elsewhere. On the 29th, the Division began to move gradually on in caterpillar fashion, drawing up a back segment to propel the forward one, inch by inch, or mile by mile. Mr. Blundell’s description in the Morning Post of the advance shows how risky and ingenuous a proceeding the movements of baggage in face of the enemy may be. “The route lay over a series of ridges and spruits and along a parallel line of hill on which the Boer forces had taken up their position. The baggage, &c., was first concentrated and taken over the spruit, with the Seaforths as right rear flank guard and the Argyll and Sutherland Highlanders as rear guard. As the baggage and transport advanced the Highland Light Infantry advanced, and the battalions guarding flank and rear retired from their position and followed the baggage across the drift, while small bodies of the enemy hovered round the retiring rear at a respectful distance and unable to do any serious damage.” Finally at 7 p.m. on the 29th, exactly to time ordered by the chief, the General and his tired warriors marched into Heilbron, having covered within eight days a distance of 126 miles, fighting “a swarm of hornets” at intervals the whole way, and losing in the advance fifty-four wounded and nine killed—a loss in comparison with the work done by no means heavy. Mr. Blundell’s description of the class of work and its reward so happily hits off the nature of the movement, that the temptation to quote him is irresistible. “To appreciate the humours of the military situation in these regions, one would have to turn to the experiences of one’s schoolboy days with wasps’ nests, when, after the capture of the main position, the survivors take to guerilla warfare in the grass, crawling up your trousers and dropping on your neck from unexpected quarters, and inflicting damage to your temper and prestige out of all proportion to the losses incurred or the advantage gained.” FROM BOSHOP TO KROONSTAD Christiana, as we know, was occupied on the 16th of May by one of General Hunter’s brigades, while Lord Methuen moved his Division from Boshop to Hoopstad, thus bringing his troops into the zone of the great operations, and pursuing his march eastwards along the south bank of the Vaal. (Hunter’s Brigade afterwards removed to cover the repair of the line along the Bechuanaland Railway towards Vryburg, and there for the present we must leave them.)
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