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# Representation and realization of bounded holomorphic functions defined on a polydomain

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Title:
Representation and realization of bounded holomorphic functions defined on a polydomain
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Book
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Tomerlin, Andrew T., 1974-
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Title Page
Page i
Dedication
Page ii
Acknowledgement
Page iii
Preface
Page iv
Page v
Page vi
Abstract
Page vii
Chapter 1. Introduction
Page 1
Page 2
Page 3
Page 4
Chapter 2. Realization and representation
Page 5
Page 6
Page 7
Page 8
Page 9
Chapter 3. Realization and representation on a polydomain
Page 10
Page 11
Page 12
Page 13
Page 14
Page 15
Page 16
Page 17
Page 18
Chapter 4. Interpolation
Page 19
Page 20
Page 21
Page 22
Chapter 5. Corona theorems
Page 23
Page 24
Page 25
Page 26
Page 27
Chapter 6. Systems theory
Page 28
Page 29
Page 30
Page 31
Page 32
Page 33
Page 34
Page 35
Page 36
Page 37
Page 38
Page 39
Page 40
Chapter 7. Conclusion
Page 41
Page 42
References
Page 43
Page 44
Biographical sketch
Page 45
Page 46
Page 47
Full Text

REPRESENTATION AND REALIZATION OF BOUNDED HOLOMORPHIC
FUNCTIONS DEFINED ON A POLYDOMAIN

By

ANDREW T. TOMERLIN

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2000

Dedicated to my parents

Dr. Arthur H. Tomerlin and Mireya M. Tomerlin

ACKNOWLEDGEMENTS

First, I would like to thank my advisor, Dr. Scott A. McCullough, for his

guidance toward the preparation of this dissertation. His insight, support, and pa-

tience are deeply appreciated. Working with such an excellent advisor was an honor

for me. I would also like to thank all of my committee members: Dr. Jacob Hammer,

Dr. Murali Rao, Dr. Li-Chien Shen, and Dr. Douglas Cenzer. Their insight and

support are also appreciated.

I would like to thank most my parents and Paromita. They have been my

inspiration, support, and life. Without them, I would not have accomplished this

work.

PREFACE

This work combines two sciences, mathematics and control theory. We assume
the reader has basic knowledge in functional and complex analysis. Some knowledge
of state variable methods would be helpful, but not necessary.
Let us define some basic terminology used throughout the text. We define a
region as an open connected subset of Cn = {(z1,... zn) : zi E C} where C is the
complex plane. A kernel defined on a region Q is a function k : Q x C. Define
the unit disk D = {z: IzI < 1} and the polydisk D' = {(z = (z,-,..., zd) zi E D}.

ACKNOWLEDGEMENTS .

PREFACE ..........

ABSTRACT .........

CHAPTERS

1 INTRODUCTION .

2 REALIZATION AND REPRESENTATION .

2.1 Representation and Realization on . .
2.2 Transfer Function Embedding Theorem .

3 REALIZATION AND REPRESENTATION ON

3.1 Realization on Q = x ... x fQd . . .
3.2 Representation on Q = 01 x ... x Qd

4 INTERPOLATION ................

4.1 Interpolation on V .............
4.2 Interpolation on f = -, x ... x Qd . . .

5 CORONA THEOREMS ..............

. . . .. . .

POLYDOMAIN

, . . . .

, o ,. .. .. .

5.1 Toeplitz Corona Theorem on . . . . . . . . .
5.2 Toeplitz Corona Theorem on Q = 91 x x . . . ...

6 SYSTEMS THEORY ...........................

6.1 Roesser Model ... ............................
6.2 Energy Conservative/Energy Dissipative Linear Miilti. lii leii -iuiali
System s . . . . . . . . . . . . . . . . .
6.3 M inimal Realizations ........................

7 CONCLUSION ..............................

REFERENCES ...................................

. . . . . . .a

... ..........

. . . . . .O . .

BIOGRAPHICAL SKETCH ..................... ....... 45

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

REPRI-:SINTATION AND REALIZATION OF BOUNDED HOLOMORPHIC
FUNCTIONS DEFINED ON A POLYDOMAIN

By

Andrew T. Tomerlin

December 2000

Chairman: Dr. Scott McCullough
Major Department: Mathematics

We define a Schur class of functions in terms of a kernel factorization property.

We characterize the elements of this Schur class as transfer functions of a certain type

of unitary colligation and as contractive functions over a class of operators acting on

a Hilbert space. With these results we establish a Nevanlinna-Pick interpolation the-

orem with our Schur class as the interpolating set. Further, we prove an extended

version of the Toeplitz corona theorem. We examine our results within a systems

theory framework and devise an efficient algorithm to realize a function of two vari-

ables.

CHAPTER 1
INTRODUCTION

An analytic function f defined on the unit disc D = {z: Izi < 1} is in H(D)
provided I|f|l = {If(z)| : z D} < oo. The study of bounded functions plays a
decisive role in electrical engineering. Rational functions H(z) in H(D) are in one
to one correspondence with linear time invariant stable physical systems. The rational
function that is identified with a particular stable system is the transfer function of
the system. Transfer functions are an input to output model of a physical system;
thus knowledge of the properties of Hco(D), in particular the unit ball in H-(D), is
important for the study of linear time invariant stable systems.
Denote the unit ball in H(D) as BH(D) = {f H(D) : I\f\l < 1}. Ele-
ments of BH(D) are transfer functions of unitary colligations. A unitary colligation
E is a tuple (U, ,N\$, E,), where W,\$, are Hilbert spaces and
[, A B]5 1 ] \ [\H]
u=C D\ C* [ [cj
is a unitary operator. The transfer function associated to E is WE(z) = D + (I -
zA)-'zB [2, 4]. A function f is in BH(D) if and only if f = WE for some unitary
colligation E.
Let s(z, w) 1 denote the Szeg6 kernel. The classical Nevanlinna-Pick
Theorem implies that a complex valued function f defined on D is in BH'(D) if and
only if the kernel Kf : D x D -+ C,
Kf(zw) = 1- f(z)f(w)(
1-z (1
is positive semidefinite. Rewriting identity (1.1), it follows that f is in BH(D) if
and only if it has the factorization property: there exists a positive kernel K such

that

1 f(z)f(w) = K(z,w)s(z,w). (1.2)

Fix a positive integer d, regions Qj, and positive kernels kj over fQj, j =
1,2,..., d, and assume that, for each j, the reciprocal of kj has one positive square,
i.e.,

-(z, w)= 1 b(z)b(w)

for some scalar valued functions b{,., b6. Let Q denote the polyregion

Qtl X Qt2 X ... x Qd

and note that the kernel k(z,w) = k1(z1,w1)k2(z2,w2)... kd(zd,Wd) is a positive
kernel over Q. Given Hilbert spaces ,,, (S,S,) denotes the space of bounded
linear operators from E to E.. Define Fk(E,E.) to be the set of C(E,,) valued
functions W(z) = W(zi,z2,... Zd) which are defined on Q and which satisfy the
factorization property: there exist auxiliary Hilbert spaces Ci,...,Cd and functions
H1,..., Hd defined on Q with values in C(Cj, .) such that
d 1
.F'Pk : I. W(z)W(w)* = zj,-( ,wj )Hj(z)Hj(w)*.

When kj = s for each j, .Fk(., S.) is known as the generalized Schur class [9] and is
denoted Fd(S, E). Elements of TFd(g, E.*) have a factorization form T.Pd:
d
.FPd : I,. W(z)W(w)* = E (zj, wj)Hj(z)Hj(w).
j=i
It is evident from .TF'Pd that J' (C, C) = BH70(Dl).
In addition to the factorization property elements of d(s, *.) and BH(D)
share two other fundamental properties. First, functions in F'd(e', .) are transfer
functions of appropriately defined unitary colligations. Define a d-variable unitary
operator colligation E to be a tuple E = (U, 9-,,), where 9-,,, are Hilbert

spaces, 7 = @3=1 Hj has a fixed d-fold orthogonal decomposition, and U is a unitary
operator
U A B]\ 'U [1-
u [C D\~ 5] '6.
The transfer function of the d-variable unitary operator colligation E is defined to be
the operator-valued function defined on 11V

Wr(z) = D + C(I- Z(z)A)-'Z(z)B

where Z : Dd -+ 7 is defined by Z(z) = dZi(z;) and Z, : D -+ 7Ui is defined by
Zi(z) = zI-H,. A function f analytic in U1 is in Td(E,E.) if and only if f = WE for
some d-variable unitary colligation E [2]. The second fundamental property Td(s, E)
shares with BH(D) is that under proper interpretation, elements of .Td(, *) are
contractive functions. A function f analytic in DI' is in .Fd(E,,) if and only if
f (Ti1, T2,..., Td) has operator norm at most one for every d-tuple of commuting strict
contractions T = (Ti,..., Td) acting on a Hilbert space H [2].
Let Q, be a region for j = 1,...,d and define Q = Ql x.. xf)d. For each region
Qj there exists an associated positive kernel kj defined on 0 such that the reciprocal
kernel has one positive square. Then Fk(S, E,) can viewed as the generalization of
Fd{(,,) from lD to the polyregion Q. Elements of Fk(E,E,) and Fd(E,E,) share
parallel properties. Elements of .Fk(C, ,) are transfer functions of an associated class
of unitary colligations. We also show elements of Yk(E, ,) are contractive functions
over an abstract class of operators acting on a Hilbert space.
The organization of the thesis is as follows: in chapter two, we examine the
results characterizing elements in rd(C, E,) [2]. We extend these results and prove
transfer functions of a certain type of contraction colligation are in Td(s, E). We
call this result the Transfer Function Embedding Theorem.
Chapter three deals with the characterization of elements in Fk(g, *). We
prove elements of Fk(8, 8) are transfer functions of an associated class of unitary

colligations. Moreover, we show that for specified kernels, elements of Fk(E, .) are

contractive functions defined over a class of operators acting on a Hilbert space.

Chapter four deals with Nevanlinna-Pick interpolation. We present the clas-

sical Nevanlinna-Pick interpolation theorem and the extension of this theorem to
several variables [1]. Using the results found in chapter three we prove an extended
Nevanlinna-Pick interpolation theorem with Fk(E, .) as the interpolating set.

Chapter five deals with the Corona problem. We begin the chapter with

a review of the Carleson Corona Theorem [11] and the Toeplitz Corona Theorem

[22, 17]. We present the Toeplitz Corona Theorem for polydisk the IV [9]. Using
the results found in chapter three we prove our own extended version of the Toeplitz

Corona Theorem for the kernel k defined on the polyregion Q = Q x ..** x fd.
In chapter six, we discuss the Transfer Function Embedding Theorem in a
systems theory framework. We show how results characterizing elements of Fd(.F, S.)

can be viewed as physical laws for multidimensional systems. Furthermore, we discuss

the minimal realization problem for functions of two variables and give an efficient
algorithm to realize a function of two variables.

CHAPTER 2
REALIZATION AND REPRESENTATION

We begin this chapter defining the Schur class Yd(e, ). This set is defined

as having a factorization property .'Pd. J. Agler [2] proved that elements of this set
have two other representations. One representation is of transfer functions of a certain
type of unitary operator. The other representation is of contractive functions over
d-tuples of commuting strict contractions acting on a Hilbert space. We present and
discuss J. Agler's results in this chapter. Furthermore, we prove transfer functions of
a certain type of contraction operator are in the set .Fd(, .).

2.1 Representation and Realization on D'

Let E be a Hilbert space and () be the space of bounded linear operators
from into . An operator valued mapping I : Q x Q -- () is positive-semidefinite
if there exists an auxiliary Hilbert space M and a function H : Q -+ 1(M,) such
that l(z,w)= H(z)H(w)*.
Let E and . be two Hilbert spaces and (, ) be the space of bounded,

linear operators from into .. Define Yd(, ) to be the set of C(, .)-valued
functions defined on IDI that have the factorization property FPd: there exist d
positive-semidefinite holomorphic operator mappings lk : D' x D -+ LC() such that
d
FPd: I W(z)W(w)* = E -(Z, wj)lj(z, w).
j=l

.Fd(\$, .) is commonly called the Schur class. Functions in Fd(, *) have
a transfer function representation. A d-variable unitary operator colligation E is a
tuple E = (U, -, , ), where '7-,, are Hilbert spaces, 7- = Eq=l Hj has a fixed

d-fold orthogonal decomposition, and U is a unitary operator
== [A B] [W _4 11
u-C D\' 9] E\
The transfer function of the d-variable unitary operator colligation E is defined to be
the operator-valued function defined on W

W (z) = D + C(I Z(z)A)-'Z(z)B

where Z U -+ 7- is defined by Z(z) = Ef= Zi(z() and Zi : D -)+ Ti is defined by
Z,(z) = ZIH,.

Theorem 2.1.1 [2] Let ,. be a Hilbert spaces. If f is a holomorphic (,E.)-
valued function defined on D, then f E Fd(S,E.) if and only if there exists a d-
variable unitary operator colligation E such that f = WE.

The representation found in Theorem 2.1.1 is a unitary realization of f. Func-
tions in Fd(E, .) also have a representation as bounded functions over d-tuples of
commuting strict contractions acting on a Hilbert space. Let T = (T.,..., Td) be any
collection of d commuting strict contractions defined on a Hilbert space H and f be
analytic (S,.)-valued function defined on VD. The operator f(T1,...,Td) can be
defined via the Cauchy integral formula having values in (F H, .F 0 H). Equiv-
alently f(T1,...,Td) can be defined via a power series expansion f(T1,...,Td) =
Zk c(k) Tjk ... Tkd where k = (k1,..., kd) ranges over Nd.

Theorem 2.1.2 [2] Let and . be Hilbert spaces. If f is a holomorphic C(,.)-
valued function defined on DJ, then f E .Fd(E,E.) if and only if IIf(T)|lop < 1 for
every d tuple T = (Ti,... ,Td) of strict commuting contractions defined on a Hilbert
space H.

The Nagy Dilation Theorem and Ando's Theorem [24] can be used to show
.Td(E, .) is identical to the unit ball BH((Dd) for d=1,2 respectively. Hence Theo-
rem 2.1.2 gives the equivalence between Fd(E, E.) and contractive analytic functions

on the disk and bidisk respectively. For d > 2, .Fd(, E.) is properly contained in
BH-(ID).

2.2 Transfer Function Embedding Theorem

Theorem 2.1.1 can be extended to transfer functions of contraction operators.
Define a d-variable contractive operator colligation E to be a tuple E = (U, 7W, ', E.),
where 7W, E, E. are Hilbert spaces, 7' = l Hj has a d-fold orthogonal decomposition,
and U is a contraction operator
[I B] :u [1 [
u~C D E] E.~
The transfer function of the d-variable contractive operator colligation E is defined
to be the operator-valued function defined on I

W (z) = D + C(I Z(z)A)-'Z(z)B

where Z : V -4+ 7 is defined by Z(z) = O-=Zi(zi) and Z, : D -+* 7'Hi is defined by
Zi(z) = zI-Hu,.

Theorem 2.2.1 (Transfer Function Embedding) If E is a d-variable contrac-
tive operator colligation, then there exists a 2d+l-variable unitary operator colligation
F such that
WVr(zl,...,z,O,...,o0) = WE(,...,Zd).

Proof: Assume that f = WE for some d-variable contraction operator colligation E.
Extend U to a unitary operator using the rotation matrix Ru [14]:

Ru [(I- U] -U : ] --+

where g is the closure of the range of (I UU*)2 and .. is the closure of the range
of (I U*U)2. Ru can be written in the following block form
= A B A, B
SC D C, D,
A2 B2 -A* -C*
(C2 D2 -B* -D*-

Using elementary column and row operations on Ru we define a unitary operator Q:
A B; A, B
C2i -D* -C* D2 ., -
A2 -B* -A* B2 "
C C C, DL-
Label
A Bi A,
A= C2 D* -C*
A2 -B* -A*

-1D21
B=2D

C= [C C2 Ci].

Using Theorem 2.1.1 and Theorem 2.1.2 we conclude

Wl(z) = W(zi,...,Zd,...,Z2d+I)

=D + 0(1 Z(z)A)-'B

is in F2d+l(, .) Let T = (T,,..., Td) be any d-tuple of commuting strict contraction
operators acting on a Hilbert space H. Computation shows WV(T1,..., Td, 0,..., 0) =
W(T1,..., Td). Hence

IIvW(Ti,...,Td, O,...,O)Ilop = IW(Ti,...,Td)IIop < 1,

since (T1,..., Td, 0,..., 0) is a 2d+ 1 tuple of commuting strict contraction operators.
Conclude
IIW(Ti,...,Td)!Iop < 1

for all d-tuples of commuting strict contractions acting on a Hilbert space H. Using
Theorem 2.1.2 conclude W TE d(, ). 0

Corollary 2.2.2 Let ,, be Hilbert spaces. If f is a holomorphic (E, )-valued
function defined on 1D0, then f E FXd(S,.,,) if and only if there exists a d-variable
contractive operator colligation E such that f = Wr.

9

Proof: If f .Fd(S,S.), then Theorem 2.1.1 shows that there exists a d-variable

contractive operator colligation E such that f = WF. The Transfer Functions Em-

bedding Theorem provides the converse. n

We call Theorem 2.2.1 the Transfer Function Embedding Theorem. This name

is appropriate due because we embed the transfer function of a d-variable contraction

operator colligation into a transfer function of a 2d + 1-variable unitary operator

colligation. We embed a transfer function by doubling the dimension.

CHAPTER 3
REALIZATION AND REPRESENTATION ON A POLYDOMAIN

In the last chapter we found that a function f .Fd(E, E.) if and only if
f = WE where E is a d-variable unitary operator colligation if and only if f is a
contractive function over d-tuples of commuting strict contractions. If we replace
each occurrence of the Szeg6 kernel in T'Pd with a suitable kernel kj defined on a
region Qj, we obtain parallel results.

3.1 Realization on Q = Qi x ... x f

For j = 1,2,... ,d, let kj(z,w) be any positive kernel defined on a region %j
for which the reciprocal kernel has one positive square, i.e.

l(zw)= > b (z)b,(w)
171
Ic 1=1

for some scalar valued functions b,3I b3 defined on Qj. The kernel defined as
k = kl(zi,wj)k2(z2,w2))... kd(Zd,Wd) is a positive kernel defined on the polyregion
Q =Q1 X ... X +d.
Given Hilbert spaces \$,., let Yk(,*) denote the set of (,E.) valued
functions W(z) = W(zI, z2,..., zd) which are defined in the polyregion Q = Q, x
* X Qd and which satisfy the factorization property FPk: there exist auxiliary Hilbert

spaces C1,... ,Cd and functions H1,..., Hd defined on Q with values in C(Cj,.) such
that
.T'Pk Ie W(z)W(w)* = (zJ,wj)HJ(z)Hj(w)*.

For kj = 1-z, the Szeg6 kernel, JFPk corresponds to the outgoing kernel
representation examined by Ball and Trent [9]. For d=l and bj(z) general for j =

1,..., L1, Fk(E, E.) corresponds to the unit ball of multipliers on the space H(k) for
the NP kernel k [10].
Let L =(L1,..., Ld) and define a d-variable L-ball operator colligation, E, to
be a tuple E = (U,H, ,.), where H,,. are Hilbert spaces, H = E)=IHj has a
fixed d-fold orthogonal decomposition, and U is a unitary operator

^-C D)\) IF.
U = ( ): (' \$.+(" ,

where H' = edL1[ HL]. Let H/ = eL 'Hj and let Zj denote the analytic function
defined on IL, with values in (H, Hj) defined by
.,ZL^): [Z IH, ... ZLIH)].

Define the polyregion BL = BL1 x . x BLd and Z : BL + C(H', H) by Z(z1,..., z') =
E4Zj (zJ). The transfer function for the colligation E is

Wi(z) = D + C(I Z(z)A)-lZ(z)B.

Define E : 0 -+ BL as the embedding of Q into BL given by

E(z) = (l,. .,Z ) = ((b1(z),... b'( )) (d(),. ( ))).

With this terminology we give the following theorem.

Theorem 3.1.1 The L(E, E.)-valued function W(z) = W(zi,...,Zd) defined on =
Q, x ... x Qd is in the class Fk(E, E.) if and only if there exists a d-variable L-ball
operator colligation E such that W = W.(E(z)).

Proof: Suppose first that W(z) = Wy(E(z)) for a d-variable L-ball operator
colligation E. Using the fact that Zj(E(zj))Zj(E(wj))* = 1 --(zj,Wj), compute

I. W(z)W(w)* = E (zj wj)Hj(z)Hj(w)
kj

where Hj(z) : Hj -+ \$ is given by Hi(z) = C(I Z(E(z))A)-'IH,.

Conversely suppose that

I. W(z)W(w)* = (zj, wj)Hj(zj)Hj(wj).

We rewrite this identity in the form

Y[Zj(E(zj))Zj(E(wj))*Hj(z)Hj(w)*] + I =
SHj(z)Hj(w)* + W(z)W(w)*. (3.1)

Let F. be the linear span of the functions
I [Z((E(wi))*Hl(w)*1
Zd(E(W dHd(w)1 e.:w=(w1,...,Wd) e, e. C(Edi(k ck))I
Zd(E(wd})*Hd(w)*
L ~I
and F be the linear span of the functions
I Hi(w)*1
~~~C (Ged1C)e.
!d(W)* e.:w= (w1,...,Wd) E Q, e .E C(LkCk)(DE.-
Hd(w)*

As a consequence of (3.1) there exists a well-defined linear isometry V from 7 onto
.'F. such that
,Hi(w)* Z,(E(wi))*HI(w)*
V: e. --+ e, (3.2)
Hd(w)* Zd(E(wd))*Hd(w)*
[W(w)* I

for all e. E F. and w E Q. Choose Hilbert spaces Hlk containing Ck such that
dim(HIk e Ck) = oc. It follows there exists a unitary map

U : (E)1 ) ( -+ (E[=(EDikH,)) E.

extending V. Set H = d!fik and write U as a 2 x 2 block matrix

A7 B]
U= [C D\

Let us set H(w) = EdHk considered as an element of C(., H). Since U extends V
from (3.2) and Z(E(w))* = EDlZk(E(wk))* we see that
[A* C*] [Z(E(w))*H(w)* e = H(w)*] e for e E
B* D* [ I IW w)*.e* for e* \$*"
This generates the following system of operator equations:

A*Z(E(w))*H(w)* + C*= H(w)*
B*Z(E(w))*H(w)* + D* = W(w)*.

From the first equation we solve for H(w)* to get H(w)* = (I A*Z(E(w))*)-IC*.
Substituting this into the second equation yields

W(w)* = B*Z(E(w))*(I A*Z(E(w))*)-C* + D*

or equivalently
W(z) = D + C(I Z(E(z))A)-
We conclude W has the desired form with U. *

3.2 Representation on fQ = f1 x ... x fd

Define the set .L(,9) to be the set of (,9)-valued functions W such
that W has the factorization property Flk with the kernels kj :BLi -+* C defined as
kj(z, w) = -<1- This kernel is commonly known as the row contraction kernel
1 --3L"
[8] and in view of the proceeding theorem, if W FL(, ), then W(z) = W.(z)
for some d variable L-Ball operator colligation. Notice, if Li = 1 for all i = 1,... ,d
we obtain -L(,.) = .Td(,). The set FL(,) is of particular importance and
we wish to characterize it in terms of bounded functions over the domain of some
class of operators acting on a Hilbert space H. Let n be a positive integer and
T = (TI,..., Tn) be an n tuple of operators acting on a Hilbert space H. We say
T = (T1,. . Tn) is a strict row contraction if 'iL Tj*T, < 1. Define the class 7R to
be the collection of d tuples of operators, T = (T1,..., Td), where T, is an Li tuple of
operators that forms a strict row contraction.

Theorem 3.2.1 IfW is a (\$, \$.)-valued holomorphic function defined on BL, then
W E .FL(,.) if and only if IIW(T)1\,p < 1 for all T R.

Let T R, then W(T) E L(\$,E.) 0L(H) is defined by

W(T) = E c(m) 0 Tm
m
where W(z) = c(m)zm, z E BL. More generally, if h is a (, E,)-valued function
defined on BL x BL holomorphic in the first variable and conjugate holomorphic in
the second variable, then h(T) E L(C, E.) 0 C(H) is defined by

h(T)= c(m,n) 0 T*n Tm
m,n
where h(z,w) = m, c(m,n)T-zm, z,w E BL. Theorem 3.1.2 follows from the
following Theorem by letting h(z,w)= Ic. W(z)W(w)*.

Theorem 3.2.2 Let C be a separable Hilbert space; if h = h(z,w) is a (C)-valued
function defined on BL x BL holomorphic in the first variable and conjugate holomor-
phic in the second variable, then h(T) > 0 for all T E 1R if and only if there exist d
auxiliary Hilbert spaces Mr and d holomorphic I(Mr,C) valued maps fr r = 1,..., d
respectively defined on BL such that
d
.FPL : h(z,w) = E (1- < Zr,Wr >CLr )fr(Z)fr(w)*
r=l
for all z,w E BL.

Proof: First assume h has the form in TPL. Fix T E R. It follows h(T) =

r=1 fr(T)(1 --iL i*)fr(T)*. Since T 7?, 1- 2LI u >u 0 and we conclude
h(T) > 0.
To prove the converse direction fix a basis {ei} of C. Let '7- denote the topo-
logical vector space of holomorphic C(C) -valued functions defined on BL x BL with
the topology induced by the family of seminorms

IlhI\n = maxzwn-g\Pnh(z,w)Pn\\

where
n-BL n -)BL,) x ... x ((n )BLd)
n+1 n+1 n + I
n~n
n--- 1- = {(Z,..., Z ) :zl E and I Z,|12 < n-f}.

Let Pn denote the orthogonal projection of C onto the span of {e, : 1 < i < n}. The
topological vector space W- carries a locally convex Hausdorff topology. Let 0 C 7-
denote the set of all h E W- such that h(z,w) = d=l( < Z,,W >CL)fr(Z,W)
where fr is positive semidefinite holomorphic on BL x AL for each r. The fact that
0 is a convex cone is easily verified.
We claim 0 is closed in X. To see this assume that
h(z,w) = E=1(- < Zr,Wr >CLr)fr(z,w) E C0 and hJ(z,w) -+ h(z,w). For n > 1
inductively construct a sequence {jj}, {jI},... as follows. Since 11hilli -+ I\h\\1, we
have
limj- 0(1- < Zr,Zr >)Plf/.(Z, )P, = Ph2(z,-)P,

for z E 'BLL. Since fr is positive semidefinite, f{(z,z) > 0, and since I\hilli forms a
bounded sequence there exists a positive bounded function g : 1BL -+ L(PC) such
that
Pi f(zz)Pi < g(z)

for all j ,r, and z IBL. Finally, since J7 is holomorphic positive semidefinite we can
conclude there exists a subsequence {j/} and d (P1C) valued positive semidefinite
holomorphic functions gr(z, w) such that

Pi/' (ZIW)Pi --+ 1

uniformly on compact subsets of !BL X !Bn.
Now suppose that sequences {j/},..., {j-}; {j/} C {jI-1}, have been de-
fined with properties
P, f (z, w)P, -- g,.(z,w)

for i = 1,..., n 1. The argument in the preceding paragraph shows that there exist
a subsequence {j'} of {jn'-} and d L(PnC) valued positive semidefinite holomorphic
functions g,(z, w) such that

Pn.fr n(z,w)Pn -+ g(Z,w)

uniformly on compact subsets of --BL x n-iL
Now define d holomorphic /2 matrix-valued functions Gr(z, w) on BL x RL by
the formula
Gr(z,w)ij = limz < gr(z,w)ej, ei >

Gr is well defined by construction: if m < n, then g-(z,w) = Pmgr(z,w)Pm. Fur-
thermore since by construction,

< h(z,w)ej,ej >= E(1- < Zr,Wr >)Gr(z,w)ij (3.3)

provided that there exist C(C) valued holomorphic maps gr with

< g(z, w)ej, ej >= Gr(z,,w) (3.4)

(i.e. Gr(z,w) is a bounded operator on 12). Since Gr(Z, z) is positive semidefinite,
(3.3) implies that Gr(z, z) is bounded. Since Gr(Z, w) is positive semidefinite Gr(Z, w)
is bounded. Hence (3.4) defines C(C) valued maps gr and we obtain from (3.3)

h(z,w) = (1- < z,w >)gr(z,w)

which establishes that 0 is closed.
Now assume that ho is a holomorphic C2(C) valued function on BL x ll with
the property that ho(T) > 0 for all T 1. Let us show ho E 0. Since 0 is closed,
the Hahn-Banach separation principle implies that ho e (09 if and only if

ReLo(ho) > 0

(3.5)

whenever Lo E 7/* has the property that

ReLo(h) > 0 for all h 0. (3.6)

Assume that L0 o 7-* and that (3.6) holds. We must show (3.5) holds. For h E W
define hV(z,w) = h(w,z)*. Define E E V* by the formula

E(h) (Lo(h) + Lo(hv)).

Let 7-o denote the vector space of holomorphic C(C,C) valued maps defined on BL.
Define a sesquilinear form [ ] on 7Ho by the formula

[f,g] = E(f(z)g(w)*).

Observe that if h 7- and h = hV then

E(h) = ReLo(h).

Hence since (f(z)f(w)*)v = f(z)f(w)* for all f E '7o we deduce from (3.6) and the
fact that f(z)f(w)* 0 that [ ]is positive semidefinite on 7o. Letting N = {f E
7W : [f, f] = 0} we deduce via Cauchy's integral formula that N is a subspace of
-Q and that [, ] induces an inner product on -'. Let '2(E) denote the Hilbert
space obtained by completing -7 with respect to this inner product. Densely define
L1 + + Ld operators T = {{Mi}'l}j=I acting on N by the formula

(M f)(z) = zif(z)
Z = (Z,..., zd) BL
zi = ( z'1 L) EL

Fix f E 2(E) and i E {1,...,d}. We have
[f 2 MiI1e
if I I2(E) II(f,...,MALf)\,I 2(E)
= E(f(z)f(w)*)- < zi, wi >CL, E(f(z)f(w)*)
= E((I- < z, w, >CL,)f(z)f(w)*) > 0,

since (1- < zi, wi >cL, )f(z)f(w)* 0. Hence T is not only well defined on N
but extends by continuity to a contraction defined on '2(E). Conclude pT R for
p < 1. Fix h = m ,, Cmn z E H, let p < 1, and let f = ej fj C -2(E).
Let us derive a formula for < h(pT)f,f >.

< h(pT)f, f >= I < Cm, (pT)*'(pT)mrf, f >
mn
= > 3 < Cmn (pT)*'(pT)m(ej 0 fj), e, 0 fi >
mn ij
= E < Cmnej, ei > [(pT)mf, (pTr)"f]
mn ij
= E < CmnCj, ei > E((-W-)n(pz)mfj(z)fi(w)*)
mn ij

= E (E < h(pz,pw)ei,ej > fj(z)fi(w)*) .
ij
Letting f = -=1 ej (lej) in the above calculation where (le,) is the natural
embedding of the mapping z -* zej E (C,C) in W-(E), we find

[h(pT),f,f] = E(Pnh(pz,pw)Pn).

Since pT E R [h0(pT), f, f] > 0 for all p < 1. Hence

E(Pnh,(pz,pw)P.) > 0

whenever p < 1 and n > 1. Fix p < 1 and let n -+* oc, by the continuity of E we
deduce that

E(ho(pz,pw) > 0 (3.7)

for all p < 1. Letting p 1- and using the continuity of E we deduce

(ho) > 0. (3.8)

Finally, since ho(T) > 0 whenever T 71 we claim that ho = h'. To see this take
T = (z1,..., Zd) E BL. In particular, we have that E(ho) = ReL(ho) and (3.8) implies
that (3.5) holds. This establishes the theorem. *

CHAPTER 4
INTERPOLATION

The classical Nevanlinna-Pick interpolation theorem states given n distinct

points zl,..., z' in D and n points y1,..., y' in C, there exists f e BH-(D) such

that f(zi) = y' for i = 1,..., n if and only if the associated Pick matrix

[(1 yiV-)s(zi, zj)i,j (4.0)

is positive-semidefinite. This theorem was extended to include the generalized Schur
class Fd(S,.) as the interpolating set by J.Agler [1]. In this chapter we present
the Nevanlinna-Pick interpolation theorem for .Fd(, E). Using the results found in
chapter three, we prove our version of the Nevanlinna-Pick interpolation theorem

with Fk(, .) as the interpolating set.

4.1 Interpolation on IID

To formulate J. Agler's Nevanlinna-Pick interpolation theorem, let z1 = (z[,
),..-, Z = (z,...,zn) be n distinct points in E#Y, M1,...,Mn be n auxiliary

Hilbert spaces, xl,..., Xn be n operators in C(., Mj) respectively, and Yi,..., yn be
n operators in (, My) respectively. The associated interpolation problem is I: Is

there a W E Y7d(, .) such that xiW(zi) = yi for i = 1,...,n.

Theorem 4.1.1 (Agler's Nevanlinna-Pick Interpolation Theorem) [2][9][3]
Let {z1,..., Zn, Xi,..., Xn, yi,. .., Yn} be an interpolation data set as shown above.
Then I has a solution if and only if there exist d positive semidefinite n x n block
matrices M' = [.1,'] such that
d

1=1

Inthecase E = = C, d= 1, and xi = 1 for i = 1,...,n it is easily seen

that Theorem 4.1.1 reduces to the classical Nevanlinna-Pick Interpolation Theorem.

For d = 2, the set Y2(C,C) is the ball in H-(IV) and hence Theorem 4.1.1 gives a

necessary and sufficient condition for interpolation in the set BH"(IV) (the unit ball
in H' (W)). For the case d > 2, .Fd(E, E.) is properly contained in the set BH'(1')
(the unit ball of the space of bounded analytic (', E.)-valued functions defined on

IDV). Thus the condition presented in Theorem 4.1.1 is in general sufficient but not

necessary for interpolation in the set BH"O(Uy).

4.2 Interpolation on f = fl x ... x d

Using results found in chapter three we can prove a Nevailinniia-Pick inter-
polation theorem with Yk(, E.) as the interpolating set. Given n distinct points
z1 = (z ,...,z ),...,z' = (zI,...,zd) in Q = Qi x ... x fd, n auxiliary Hilbert

spaces M1,..., Mn, n operators X1,..., Xn in C(., Mj) respectively, and n opera-
tors yl,... Y, in C(E, Mj) respectively. The associated interpolation problem is 1:

Find a W E Fk(S, S.) such that xiW(zi) = yi for i = 1,..., n.

Theorem 4.2.1 Let {z,... "a,...,Xn,yl,... ,yn} be a interpolation data set as
shown above. Then Z has a solution if and only if there exist d positive semidefinite

n x n block matrices M' = [Mj] such that
d
xix --(Z, z)M (4.1)

Proof: Suppose that W C Fk(E, \$.) satisfies the interpolation condition 1.
Hence by definition we know that there exists holomorphic functions Hj(z) such that

Ie. W W(z)W(zJ)*= T (z', z)Hi(zi)H(zJ)*.

Hence
d
xi(I". W(zi)W(zJ)*)x = 1 -(z, z/)xi(z)H(zJ)* x.
1-=1I

Using the interpolation condition (xiW(z') = yi) we see (4.1) holds with
M ~j =X i HI( '.)H Z j* *
M= xiHI(zi)Hl(z)*x].

If we let M' be the block matrix [MLj] it is clear from the form of the MLj that M' is
positive semidefinite.
Now suppose that there exists positive semidefinite matrices M1,..., M' for
which (4.1) holds. As each M' is positive semidefinite, we may factor M1 as M- =
A'(A')* where A' = (Ci, GinMi) for an auxiliary Hilbert space C\, = 1,..., d, and
let
A' = : where A' E C(C, Mi).

Using the fact that Zj(E(zj))Zj(E(wj))* = 1 v(zjwj), we can rewrite (4.1) as
d d
E[Z,(E(z))Z,(E(zj))A(A')*] + xxj = Ai(A.)* + yyj. (4.2)
/: 1 j l
1=1 =
Let (, be the span of the set of elements
{ Z1(E(z'))*(A) *
I
C ((dl (,Lk Ck)

Zd(E(z' (Ad). mj: mj EMj,j =1,...,n C kl
x 1
And let g be the span of the set of elements

^)*
L Yj J
As a consequence of (4.2) there exists a well-defined linear isometry V from g onto
Q such that
S(A)*/ z,(E(zi)*(A)*
V I Zd(E(z )(A)]* mr (4.3)
3 d
AjL X!}

for all mj E Mj and j = 1,..., n. Choose Hilbert spaces Hk containing Ck such that
dim(Hk 0 Ck) = oo. It follows there exists a unitary
d -d L/_
u : (=1 Hk) e E (E_(1 kHk)) E '.

extending V. Set fH = EHk, write U in 2 x 2 block operator notation,
and set Z(E(z)) = EdZ,(E(zi)) for z = (Z1,... Zd) Q. Define W(z) = D + C(I-
Z(E(z))A)-'Z(E(z))B. Then W E .Tk(, .) by Theorem 3.1.1. To show W satisfies
the interpolation condition Z we proceed in a manner similar to the last part of the
proof of Theorem 3.1.1. For a fixed j = 1,..., n let Hj be the operator

Hj= =
F(A. )*1

considered as an element of C(Mj, H). Since U extends V we know by (4.3) that

A*Z(E(zJ))*Hj + C*x* = Hj
B*Z(E(zJ))*Hj + D*x = yj.

Solve the first equation to obtain

Hj = (I A*Z(E(zJ))*)-lC*xj.

Plugging this into the second equation we obtain

(B*Z(E(zJ))*(I A*Z(E(zJ))*)-1C* + D*)x3 = yj.

Taking the adjoint of both sides we obtain the desired result xjW(zj) = yj. U

CHAPTER 5
CORONA THEOREMS

Suppose that a,,..., an are complex valued functions in H "(D). The Carleson

Corona Theorem [11] asserts the existence of fl,..., fn solving E', fiai = 1 if and

only if there exists a 5 > 0 such that

infij J > 0.

The Toeplitz Corona Theorem [22, 17] states that there exist functions fi E
n
H(l,) for i = 1,...,d such that =U fiai = 1 and sup\.<1{ZI,= Jfi(z)[2} < if

and only if
1
TaTal +. + TTa2 5-7I > 0 (5.1)

where Tak : h(z) -+ ak(z)h(z) is the analytic Toeplitz operator on the Hardy space

H2(ID) with symbol ak. Condition (5.1) can be expressed as

E al(zi)al(zj) + - + an(zi)an(zj) J2_
-1 ~ 1 2;z,:~
ij=l

for all complex numbers c1,..., CN and all points z1,... ,zN E D for N = 1,2,3,....
In this chapter we present two versions of the Toeplitz Corona Theorem. The

first version was formulated for functions defined on the polydisk D'1 [9]. Using the

results in chapter three we prove the second version for functions defined on the
polyregion Q.

5.1 Toeplitz Corona Theorem on D'

Ball and Trent [9] formulated a version of the Toeplitz Corona Theorem for

the polydisk DI.

Theorem 5.1.1 [9] Let a1,..., an be complex valued functions in H(D) and let S
be a positive number. There exist functions fi,..., fn such that the column matrix
function [fi ... f"]T is in the set !Ed(C,C") and ai(z)fi(z) + .. + a(z)fn(z) = 1
on Dd if and only if there exist auxiliary Hilbert spaces C\,..., Cd and d holomorphic
functions Hl(z),...,Hd(z) on Vd, with Hk(z) having values in (Ck,C) such that
n d
Y ak(z)ak(w) 82 = (zk, Wk)Hk(z)Hk(W)
k=l k=l
for all z,w E Dd.

The techniques found in Ball and Trent lead them to a more general theorem.

Theorem 5.1.2 Let 1,2,E3 be three Hilbert spaces and suppose that A and B are
given bounded holomorphic functions on Vd with values in (2,3) and C(1,3)
respectively. Then there exists a F e Fd(1, 2) with A(z)F(z) = B(z) on D' if and
only if there exist d auxiliary Hilbert spaces C1,. . ,Cd and d holomorphic functions
Hl(z),...,Hd(z) on V1, with Hk(z) having values in C(Ck, 3) fork= 1,...,d, such
that
d
A(z)A(w)* B(z)B(w)* = W (zk wk)Hk(z)Hk(w)*
k=
for all z,w DW.

Notice we recover Theorem 5.1.1 from Theorem 5.1.2 by taking E = 2 = C,
2 = Cn, A(z) = [a,(z) ... an(z)], and B(z) = S.

5.2 Toeplitz Corona Theorem on 0 = Qi x ... x i.

Let and . be two Hilbert spaces, kl,..., kd be positive kernels defined on
fj whose reciprocal has only one positive square. Using results found in chapter
three we prove an operator version of the Toeplitz Corona Theorem for the kernel
k(z, w) = ki(zi, wi)... kd(Zd, Wd) defined on the polyregion Q = Q, x .. x Qd.

Theorem 5.2.1 Let .1, 2,3 be three Hilbert spaces and suppose that A and B are
given bounded holomorphic functions on Q with values in .(2, E3) and (1, E3) re-
spectively. Then there exists a FE .Fk(1,2) with A(z)F(z) = B(z) on Q if and only
if there exist d auxiliary Hilbert spaces C1i,... ,Cd and (Ck, 3)-valued holomorphic
functions Hk(z) defined on Q for k = 1,..., d such that
d
A(z)A(w)*- B(z)B(w) = (zi,wi)H,(z)Hi(w)
1=1
for all z,w E Q.

Notice we recover Theorem 5.1.2 from Theorem 5.2.1 by letting E = 3 = 3
62 = e1, A(z) = [a,(z) ... a,(z)], B(z) = al., nd ki = s for 1 = 1,...,d.

Proof: Suppose that there exists a F(z) E 1k(E1,E2) such that A(z)F(z) =
B(z) on Q. Using Theorem 3.1.1 we know that there exists a d-variable L-Ball
operator colligation E such that F(z) = Wr(E(z)), where z -* E(z) is the embedding
of Q into BL1 x ... x BLd defined in Theorem 3.1.1. From A(z)F(z) = B(z) and using
the fact Zj(E(zj))Zj(E(wj))* 1 -(zjWj) we deduce

A(z)A(w)* B(z)B(w)* = A(z)(I F(z)F(w)*)A(w)*

> jj(zj, wj)Hj(z)Hj(w)*

where Hij(z): Hj -+ 3 is given by Hi(z) = A(z)C(I Z(E(z))A)-1lHj.
Conversely suppose that
d
A(z)A(w)*- B(z)B(w)*= V (z, wi)Ht(z)Hi(w).
kll

We rewrite this identity in the form

Y[Zj(E(zj))Zj(E(wj))*Hj(z)Hj(w)*] + A(z)A(w)* =

SHg(z)Hj(w)* + B(z)B(w)*. (5.2)

Let .FZ be the linear span of the functions
I Z( (E(wi)) H1(w)*
Zd(E(Wd))*Hd(w) e:W = (w,..., Wd) e, C (L))
L A(w)*
and F be the linear span of the functions
[ H, (w)
I IC (E~d~1Ck) EDE.
SHd(w) e.:w=(wi,...,w), eflE, C(Le* E)
IB(w)*
As a consequence of (5.2) there exists a well-defined linear isometry V from .7 onto
.T such that
"Hl(w)*' [Z(E(w,))*H(w)*
V : I e* -C (5.3)
SHd(w)* Zd(E(wd))*Hd(w)*
B(w)* A(w)*

for all e E E. Choose Hilbert spaces lk containing Ck such that dim(HkeCk) = oc.
It follows there exists a unitary map
d d (E),(Lk&)E
U: (k=iHk) ( E -+ (eL(E 1Hk)) E

extending V. Set f = ( fIk and write U as a 2 x 2 block matrix

U-C B]

Let us set H(w) = EHk(w) considered as an element of (S3, H). Since U extends
V from (5.3) and Z(E(w))* = edfZk(E(wk))* we see that
[A: C*] [Z(E(w))*H(wY] C = -H~w) *fr e
B* D"J A(w)* = B(w) efor e .

This generates the following system of operator equations:
A*Z(E(w))*H(w)* + C*A(w)* = H(w)*
B*Z(E(w))*H(w)* + D'A(w)* = B(w)*.

From the first equation we solve for H(w)* to yield

H(w = (I A*Z(E(w)))1C*A(w)*.

Substituting this into the second equation yields

B(w)* = B*Z(E(w))*(I A*Z(E(w))*)-lC*A(w)* + D*A(w)*

= (B*Z(E(w))*(I A*Z(E(w))*)-lC* + D*)A(w)*

or equivalently

B(z) = A(z)F(z) (5.4)

where F(z) = D + C(I Z(E(z))A)-'Z(E(z))B. Using Theorem 3.1.1 we know
F Fk(a1,E2) and (5.4) gives us our desired result. 0

CHAPTER 6
SYSTEMS THEORY
6.1 Roesser Model

Define a d-variable operator colligation E to be a tuple E = (U, 7, E, S.), where
7-,\$, \$ are Hilbert spaces, 7h = =iHj has a fixed d-fold orthogonal decomposition,
and U is a bounded operator U : 'W @ -F 7'H \$ .

u == [A B\ **e H (6.1)

7'H is known as the state space, E is known as the input space, E" is known as the
output space, and U is known as the connecting operator. As we have mentioned
before, the colligation is unitary or contractive according to whether the connecting
operator is unitary or contractive. The transfer function of the d-variable operator
colligation E is defined to be the operator-valued function defined on d

Ws(z) = D + C(I Z(z)A)-lZ(z)B

where Z W -* 7- is defined by Z(z) = djZi(zi) and Z, : D) -+ Hi is defined by
Zi(z) = zI.
Associated with any d-variable operator colligation is a d-dimensional discrete-
time linear system. The time variable n for this system is a d -tuple n = (ni,..., rid)
of integers. Define Uk : Zd + Zd to be the forward shift in the kth coordinate

7k(nl,..., rid) = (n1,... nk + 1, ... rid).

The input-state-output linear system associated with (6.1) can be written as the
following system of equations
[A B] [x(n)] = [x(a(n)) (6.2)
[C D\ u(n) y y(n)

xi(n)
where u(n) is the input signal with values in , x(n) = : is the state vec-
[Xd(n)_
xl((ui(n))
tor with xk having values in U-k, x(a(n)) = and y(n) is the output
X d ((d(n))_
signal with values in E,. Form (6.2) is referred to as the Roesser Model [15, 16] in
multidimensional systems theory.

Let us motivate why the Roesser Model is studied. The input to output

characteristics of a multidimensional linear time invariant system is modeled by a

multivariable function H(zi,..., Zd). This function is commonly called the transfer
function of the system (this motivated our use of this terminology earlier). If we
assume H(z) is a analytic function in some region Q C Cd which includes the origin
(0,..., 0) then H(z) has a power series expansion
00
H(z)= ) c(k)z (6.3)
*' k=(0...,0)

valid in 0 where k =(kl,...,kd), ki Z+, for i = 1,...,d and zk ...Z
Physically, the indeterminates z1,..., Zd are the respective delay variables along the

spatial or temporal directions of sampling during analog to digital conversion of a
multidimensional spatio-temporal signal. Functions of the form found in (6.3) are
commonly called d-dimensional filters. A realization of H(z) is a d-variable operator
colligation E generating a linear system of equations found in (6.1) and (6.2) with
WF,(z) = H(z). Thus a Roesser Model is a input-state-output model of a realization
E of a d-dimensional filter WE(z).

We discuss two aspects of multidimensional systems: (1) the physical inter-
pretation of the unitary or contractive properties of U as a energy conserving/energy
dissipative property and (2) the realization and dimension of the state space 'HI of a
realization for a real valued rational function H(z) in 2 variables.

6.2 Energy Conservative/Energy Dissipative Linear Multidimensional Systems

The results of chapter two show that the elements of the set .Td(, .) are

transfer functions of d-variable contractive and/or unitary operator colligations. Let
f : D' - (, .) be analytic and suppose there exists a d-variable unitary opera-

tor colligation E = (U, 7-,, E,.) such that f = Wr. Since U is isometric it follows

I\U(h,e)I\ = IJ(h, e)II for all (h,e) E W D . In particular Ilh|2 I+ I ell2 = IU(h,e)1\2.
Physically, this is a energy conserving property. On the other hand, let f : W -+

(, .) be analytic and suppose there exists a d-variable contraction operator col-
ligation E = (U, 7-,, .) such that f = WE. Since U is a contraction it follows

I|U(h,e)ll < I I(h,e) 1 for all (h,e) E 'H E. In particular IIU(h,e)||2 _< I|lhI2 + I|ell2.
Physically, this is a energy dissipative property.

Thus functions in Fd(, E) have realizations in terms of operators that have
either energy dissipative and/or energy conserving properties. Moreover, functions
that have either a energy dissipative and/or energy conserving realizations are in

.Fd(, .). This gives the results found in chapter two physical meaning. In partic-
ular the Transfer Function Embedding Theorem show that we can embed a transfer

function of a d-variable contraction operator colligation in a transfer function of a

2d + 1-variable unitary operator colligation. Physically this can be viewed as a law
of entropy for transfer functions. In particular, a energy dissipative system can be
embedded into a larger energy conserving system. In this way, the results of chapter

two can be viewed as empirical laws for multidimensional systems.

6.3 Minimal Realizations

We call the dimension of the state space the order of the realization. Given
H(zi, z2) a rational 2-dimensional filter, the minimal realization problem asks does
there exist a method to find a realization E of H such that the dimension of the state

space 7- is as small as possible, hence there exists no other realization A of H that
has a smaller order.
A method to develop such a realization in general is unresolved in the open
literature, but methods exist for special cases: a) 2-d transfer functions with a sep-
arable numerator or denominator [18]; b) 2-d, 3-d, and N-d transfer functions that
can be expanded into a continued fraction expansion [21, 19, 6, 20, 5]; c) 2-d all-pole
and all-zero transfer functions [25].
To keep the notation of the literature, let us replace the indeterminates z1, z2
with z', z21 representing the delay elements. A general method producing a low
order, but not necessarily minimum order, realization for a 2-d transfer function

H(zi, z2) was given in [18] in terms of hardware design of the delay elements z, z2 .
To begin, we write the transfer function H(zi, z2) in terms of a rational function of
z-1 with polynomial coefficients in z,1

Z 2 1f/(Zr1) + ** + Z2 Vi(Z11) + fo(Zi1) /
H(z1,z2) = z2M2 qmf2(Z1I) + + z2 ZI(z ') + fo(zT-) (6.3)
H~zl z2! 2qmw(Z1) +...* + zfq(z'i1) + q0(Zl-1)'

Without loss of generality, we can assume qo(z1) = 1 + 4Qo(z1). Let n to be the
maximum degree of the polynomials fmi,... fo and qm ,..., q0. Then we can write

fi(zT1) = = o fkzik and qj(zi1) = o qzk for i = 0,...,mi and j= 0,..., m2.
The realization of H(zi,z2) is shown in Fig. 6.1 and consists of three arrays of
delay elements. One array consists of n elements of type zi1 are commonly shared
to realize both numerator and denominator. The two other arrays consist of mi
and m2 elements of type z21 that feedforward and feedback to realize numerator and
denominator respectively. The order of the realization is n+m1+m2. This realization
technique is a 2-d extension of the controller canonical form where the gains of the
multipliers are functions in z21. The drawback with this realization technique is
that the feedforward delay elements of type z21 and the feedback delay elements of
type z21 are not shared. Indeed, this realization technique will not give the minimal
realization in all examples. We will discuss examples later.

u

0[qo q'I
f a f

1 Lc k

y

- ml
Z,'
delays

f n
z
Delays
qIni

S m2

delays

q.

Figure 6.1: n + m, + m2 -Order Realization[18]

As we have mentioned, the realization technique found in [18] will not always

produce the minimal realization of a 2-d transfer function. The drawback is that the

delay elements of type z2 1 are not shared. To improve this method, let us devise

a method of sharing the delay elements z21. Begin by writing down the transfer

function H(zi,z2) as shown in (6.3). Without loss of generality assume mI = mi2.

Then (6.3) induces two vectors IV, 'D with polynomial entries
fm, (.Z11)

N = (6.4)
f/o(zI1)
[qm i(z1)
2 = (6.5)
qo(zr')

where KV denotes the coefficients of the numerator of H and D9 denotes the coefficients

of the denominator of H. Define the shift S of a column vector x = as

x1'
X"l

Sk = Xk

XX\

X1
Pk Xk=

0

To share delay elements of type z21 in the feedforward and feedback arrays

shown in Fig. 6.1 we must find constants a,,... ak such that

Pk(D) = al'Pk(W) + a2Pk(S(K)) + **- + akPk(S(k-(A))

(6.6)

fo f, f

a ,, f, f* ml
a, Z2
delays

f.-n .1r-0n ^^ -7
u

:-'() 1 0 '{_ ] 0,'""' b +`t' .., + ..................................... C-0 l A.l- O L = oJ, + --0 de ysz,
-LJi-' k delays
ig r 2 1 2 +k A -

a2-1 A .2-l A W2- k WA A i A A
-.- m2-k
z2
1 1 1 delays

Figure 6.2: n + m, + m2 k -Order Realization

for some k = 1,2,.... In other words, the entries of D must be eliminated by the
span of copies of the shifts of JK. For each positive k for which (6.6) holds we feedback

the feedforward delay elements of Fig. 6.1 as shown in Fig. 6.2. We bookkeep the

sum of the feedback gains during this process and we subtract this sum from the

denominator. We obtain the following matrix equation:

0

k
,T _-Y ai^-l^ .= 0
D atm'- k(Z)

1 (z- 1)

where /i(z-1) = -= lzjk for i = 0,... ,m2 k. We then realize this difference

using m2 k delay elements of type z21 as shown in Fig. 6.2. The order of the

realization is n + m, + m2 k. The same process can be applied with the roles of

V and A[ reversed. The difference is we feedforward the feedback delay elements of

Fig. 6.1 in a similar method as shown in Fig 6.2 to produce a realization of order

n + mi + m2 k.
Starting from a transfer function H(zi, z2), our method provides a low order

realization of H. A natural question arises: given a realization E of H can we verify
that E is a minimal realization of H? This question is unresolved in the literature,

nevertheless, we can derive a method that reduces the order of a realization.
Let E = (U,`/,,E) be a d-variable operator colligation. Let Pk : 7" -+ Hk

denote the orthogonal projection. The colligation E is closely inner connected if
the smallest subspace invariant for A,P1,..., Pd and containing im B is the whole
space X. Similarly, E is closely outer connected if the smallest subspace invariant

for A*,Pi,...,Pd and containing im C* is the whole space W. For one variable

d = 1, closely inner connectivity is controllability. Similarly, for d = 1, closely outer

connectivity is observability. Thus closely inner and outer connectivity can be viewed
as extensions of controllability and observability. Ball and Trent [9] discuss closely
inner connected and closely outer connected colligations in the context of isometric,
coisometric, and unitary operator colligations. We will discuss these ideas in the

context of reducing the state space of a realization.

In [9], Ball and Trent point out that, in general, given a d-variable operator
colligation E = (U = [A ] ,,,), the compressed colligation

Eo = (U [PoA P0B \ues,71oEe)

where 7Wo C 7W is the smallest subspace invariant for A,P1,..., Pd and containing im
B (closely inner connected), and P0 : W -+ 7"o is the orthogonal projection, retains
the same transfer function. In other words, W, = W.0. Similarly, if No is the smallest
subspace invariant for A*,Pi,..., Pd and containing im C* (closely outer connected)

then WE = WE0. Hence if W is not both closely inner connected and closely outer
connected, then we can project down to a smaller invariant subspace and obtain a

realization of smaller order. This method combined with our previous results gives
us an algorithm to obtain a low order realization of a 2-d transfer function H(zi, z2).
ALGORITHM:
1. Write H(zi, z2) in the form given in (6.3).
2. Realize H according to Fig. 6.1.

3. Write down the numerator and denominator vectors A/ and 7D.

4. Check if equation (6.6) holds for any positive integer k. If so, then realize
H according to Fig. 6.2. If not, keep realization found in step 2.
5. Label delay elements and write down the matrix U = [ DB].

6. Check to see if the realization is closely outer or closely inner connected. If
so, stop. If not, then project to smaller invariant subspace and repeat step 5.
EXAMPLE 1: Let Hl(zl,z2)= z'2-. We write H(zli,z2) = z+-'i z. Ap-

plying the method presented by [18] it is easily seen that H1 has a order 3 realization
with circuit diagram shown in Fig. 6.3. Let us apply our technique. Writing the
numerator and denominator vectors we obtain

D [i'] (6.7)

Notice the first entry of N" is 1 while the first element of D is -zl1. These two
elements are linearly independent, thus we cannot find scalars a1,... ak such that
(6.6) will hold for any k. Hence we cannot use feedback to produce a lower order
realization.

0

z2

x3

Figure 6.3: Order 3 Realization of Hi

Writing down A, B, C, and D:
0 0 -
A= 00 0

0 0
C= 1 1 0]

D=0.

The orthogonal projections P1 and P2 correspond to the labels on the delay elements

in Fig. 6.3. In other words P1 corresponds to the state x, hence it is the projection

of C' onto the first coordinate. Similarly, P2 corresponds to x2 and x3, hence it is

the projection of C' onto the second and third coordinates. It is easily verified that

this realization is closely inner and outer connected. According to our algorithm, we

stop. Conclude H1 has a order 3 realization. Indeed it is proved in [18] that 3 is the

order of a minimal realization of H1.

--2 --2
EXAMPLE 2: Let H2(Zl, Z2) Z 2 Applying the method pre-
i+ 2 Z2 +--l Z2
sented by [18] it is easily seen that H2 has a order 6 realization with circuit diagram

shown in Fig. 6.4. Let us apply our technique. Writing the numerator and denomi-

nator vectors we obtain

V= [0
0

)= zl] (6.8)

Notice that PID = PiN. We feedback according to Fig 6.2 to obtain a realization

of order 5 as shown in Fig. 6.5.

Writing down A, B, C, and D:

0 0 -1 0 -1
1 0 0 0 0
A= 1 0 0 0 0
0 1 0 0 0
0 0 0 1 0 J
~1
0
B= 0
0
0
c=[0 0 0 0 1]

D=0.

It is easily verified that this realization is closely inner and outer connected. Accord-

ing to our algorithm, we stop. We conclude H2 has a order 5 realization.

To demonstrate how powerful the idea of closely inner and outer connectivity

are let us show that we could have projected down the order 6 realization of H2

shown in Fig 6.4 to obtain a order 5 realization. Writing down A, B, C, and D for

the realization shown in Fig 6.4 we obtain:

0 0 0 0 0 -1
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 1 0 0 0 0
1 0 0 0 1 0
1
0
0
B 0
0
0

C= [0 0 0 1 0 0]

D=0.

This realization is not closely inner connected. The smallest subspace invariant for

A,P1, and P2 and containing im B is the space spanned by {e1, e2, !(e3 + e5), e4, e6}

where ei is the column vector consisting of all zeros except for 1 in the ith entry.

Projecting down on this invariant subspace we obtain a realization of order 5 with

A, B, C, and D:

-0 0 0 0 -1-
1 0 0 0 0
A= 0 V'2 0 0 0
0 0 0 0
1 0 0 0
1
0
B= 0
0
0
C= [0 0 0 1 0]

D= 0.

It is easily verified that this new realization is both closely inner and outer connected.

40

4 y

0 X
-

z 0
X,'
x

Xx5

x6

X,

Figure 6.4: Order 6 Realization of H2
5

0 y

ox
-1

0
U---

Figure 6.5: Order 5 Realization of H2

CHAPTER 7
CONCLUSION

In chapter two we defined the Schur class Fd(S, E). The results of J. Agler

[2] showed elements of the set Fd(E, E) have two other equivalent representations.

One representation is of transfer functions of d-variable unitary operator colligations.
The other representation is of contractive functions over d-tuples of commuting strict

contractions acting on a Hilbert space. Also included in chapter two was the Transfer

Function Embedding Theorem. The Transfer Function Embedding Theorem can be

viewed both mathematically and physically. Mathematically, transfer functions of d-

variable contraction operator colligations are in the Schur class Fd(E, E,). Physically,

it is a law of entropy.

In chapter three, we generalized the results of J.Agler found in the previous

chapter to Fk(E, .). Elements of Fk(E, E,) are transfer functions of d-variable L-

ball operator colligations. Moreover, elements of Fk(E, E.) are contractive functions

defined over a class of operators 7R acting on a Hilbert space. Using the results in

chapter three we proved versions of the Nevanlinna-Pick interpolation theorem and

the Toeplitz corona theorem found in chapters four and five respectively.

In chapter six, we discussed the minimal realization problem for a 2-d transfer

function. As we have mentioned, this problem is still unresolved in the open literature.
Moreover, due to the complexity of the problem, it is not a subject of active research

today. Indeed, most of the research on this problem was done over two decades
ago [18, 15, 16]. Hopefully, the methods and ideas found in this dissertation will

lead to new insight on the problem. In particular, how are the ideas of inner and

outer connectivity related to the minimal realization? We know we can reduce the

42

order of a realization using inner and outer connectivity, but when does this method

fail? Hopefully, these questions will lead to answers and new insights concerning the

minimal realization problem.

REFERENCES

[1] J. Agler, Interpolation, Unpublished.
[2] J. Agler, On the representation of certain holomorphic functions defined on the
polydisk. Topics in Operator Theory: Ernst D. Hellinger Memorial Volume,
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BIOGRAPHICAL SKETCH

Andrew T. Tomerlin was born in Orlando, Florida, on July 25, 1974. He

graduated from the University of Florida in August 1996 with a Bachelor of Science

with Honors in physics. In December 2000 he will be graduating with his Ph.D in

mathematics and a MS in electrical engineering. While at the University of Florida

he met and married Paromita Bose from New Delhi, India.

I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
/U tT- 1 < 'J//'P
Scott McCullough Chairman
Associate Professor of Mathematics

I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.

Murali Rao
Professor of Mathematics

I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.

Li-Chien Shen
Professor of Mathematics

I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.

Douglas e6nzer
Professor of Mathematics

I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Docto h sophy.

a ob Haviler
Prfessor of Electrical Engineering

This dissertation was submitted to the Graduate Faculty of the Department of
Mathematics in the College of Liberal Arts and Sciences and to the Graduate School
and was accepted as partial fulfillment of the requirements for the degree of Doctor
of Philosophy.

December 2000 ___________

ri0
ILD

UNIVERSITY OF FLORIDA
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