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Page i Dedication Page ii Acknowledgement Page iii Preface Page iv Table of Contents 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 
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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. LiChien 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}. TABLE OF CONTENTS 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 NevanlinnaPick 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 Ifl = {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 NevanlinnaPick 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)( 1z (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 dvariable unitary operator colligation E to be a tuple E = (U, 9,,), where 9,,, are Hilbert spaces, 7 = @3=1 Hj has a fixed dfold orthogonal decomposition, and U is a unitary operator U A B]\ 'U [1 u [C D\~ 5] '6. The transfer function of the dvariable unitary operator colligation E is defined to be the operatorvalued 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) = zIH,. A function f analytic in U1 is in Td(E,E.) if and only if f = WE for some dvariable 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 dtuple 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 NevanlinnaPick interpolation. We present the clas sical NevanlinnaPick interpolation theorem and the extension of this theorem to several variables [1]. Using the results found in chapter three we prove an extended NevanlinnaPick 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 dtuples 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 positivesemidefinite 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 positivesemidefinite 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 dvariable unitary operator colligation E is a tuple E = (U, , , ), where '7,, are Hilbert spaces, 7 = Eq=l Hj has a fixed dfold orthogonal decomposition, and U is a unitary operator == [A B] [W _4 11 uC D\' 9] E\ The transfer function of the dvariable unitary operator colligation E is defined to be the operatorvalued 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 dtuples 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 dvariable contractive operator colligation E to be a tuple E = (U, 7W, ', E.), where 7W, E, E. are Hilbert spaces, 7' = l Hj has a dfold orthogonal decomposition, and U is a contraction operator [I B] :u [1 [ u~C D E] E.~ The transfer function of the dvariable contractive operator colligation E is defined to be the operatorvalued 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) = zIHu,. Theorem 2.2.1 (Transfer Function Embedding) If E is a dvariable contrac tive operator colligation, then there exists a 2d+lvariable unitary operator colligation F such that WVr(zl,...,z,O,...,o0) = WE(,...,Zd). Proof: Assume that f = WE for some dvariable 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 dtuple 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 dtuples 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 dvariable 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 dvariable 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 dvariable contraction operator colligation into a transfer function of a 2d + 1variable 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 dvariable unitary operator colligation if and only if f is a contractive function over dtuples 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 = 1z, 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 dvariable Lball operator colligation, E, to be a tuple E = (U,H, ,.), where H,,. are Hilbert spaces, H = E)=IHj has a fixed dfold 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 dvariable Lball operator colligation E such that W = W.(E(z)). Proof: Suppose first that W(z) = Wy(E(z)) for a dvariable Lball 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 welldefined 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* $*" This generates the following system of operator equations: A*Z(E(w))*H(w)* + C*= H(w)* B*Z(E(w))*H(w)* + D* = W(w)*. From the first equation we solve for H(w)* to get H(w)* = (I A*Z(E(w))*)IC*. Substituting this into the second equation yields W(w)* = B*Z(E(w))*(I A*Z(E(w))*)C* + D* or equivalently W(z) = D + C(I Z(E(z))A) We conclude W has the desired form with U. * 3.2 Representation on fQ = f1 x ... x fd Define the set .L(,9) to be the set of (,9)valued functions W such that W has the factorization property Flk with the kernels kj :BLi +* C defined as kj(z, w) = <1 This kernel is commonly known as the row contraction kernel 1  [8] and in view of the proceeding theorem, if W FL(, ), then W(z) = W.(z) for some d variable LBall operator colligation. Notice, if Li = 1 for all i = 1,... ,d we obtain L(,.) = .Td(,). The set FL(,) is of particular importance and we wish to characterize it in terms of bounded functions over the domain of some class of operators acting on a Hilbert space H. Let n be a positive integer and T = (TI,..., Tn) be an n tuple of operators acting on a Hilbert space H. We say T = (T1,. . Tn) is a strict row contraction if 'iL Tj*T, < 1. Define the class 7R to be the collection of d tuples of operators, T = (T1,..., Td), where T, is an Li tuple of operators that forms a strict row contraction. Theorem 3.2.1 IfW is a ($, $.)valued holomorphic function defined on BL, then W E .FL(,.) if and only if IIW(T)1\,p < 1 for all T R. Let T R, then W(T) E L($,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)Tzm, 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 = maxzwng\Pnh(z,w)Pn\\ where nBL n )BL,) x ... x ((n )BLd) n+1 n+1 n + I n~n n 1 = {(Z,..., Z ) :zl E and I Z,12 < nf}. 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 {jI1}, 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 niL Now define d holomorphic /2 matrixvalued 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 HahnBanach 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 7o 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 NevanlinnaPick 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 positivesemidefinite. 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 NevanlinnaPick interpolation theorem for .Fd(, E). Using the results found in chapter three, we prove our version of the NevanlinnaPick interpolation theorem with Fk(, .) as the interpolating set. 4.1 Interpolation on IID To formulate J. Agler's NevanlinnaPick 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 NevanlinnaPick 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 NevanlinnaPick 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 NevailinniiaPick 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 welldefined 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 57I > 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 dvariable LBall 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 welldefined 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 UC 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 dvariable operator colligation E to be a tuple E = (U, 7, E, S.), where 7,$, $ are Hilbert spaces, 7h = =iHj has a fixed dfold 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 dvariable operator colligation E is defined to be the operatorvalued 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 dvariable operator colligation is a ddimensional 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 inputstateoutput 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 Uk, 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 spatiotemporal signal. Functions of the form found in (6.3) are commonly called ddimensional filters. A realization of H(z) is a dvariable 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 inputstateoutput model of a realization E of a ddimensional 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 dvariable contractive and/or unitary operator colligations. Let f : D'  (, .) be analytic and suppose there exists a dvariable 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 Ilh2 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 dvariable contraction operator col ligation E = (U, 7,, .) such that f = WE. Since U is a contraction it follows IU(h,e)ll < I I(h,e) 1 for all (h,e) E 'H E. In particular IIU(h,e)2 _< IlhI2 + Iell2. 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 dvariable contraction operator colligation in a transfer function of a 2d + 1variable 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 2dimensional 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) 2d transfer functions with a sep arable numerator or denominator [18]; b) 2d, 3d, and Nd transfer functions that can be expanded into a continued fraction expansion [21, 19, 6, 20, 5]; c) 2d allpole and allzero 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 2d 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 z1 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(Zl1)' 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 2d 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 2d 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 .1r0n ^^ 7 u :'() 1 0 '{_ ] 0,'""' b +`t' .., + ..................................... C0 l A.l O L = oJ, + 0 de ysz, LJi' k delays ig r 2 1 2 +k A  a21 A .2l A W2 k WA A i A A . m2k 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(z1) = = 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 dvariable 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 dvariable 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 2d 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 dvariable unitary operator colligations. The other representation is of contractive functions over dtuples 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 dvariable 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 NevanlinnaPick 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 2d 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. 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Kailath New Results in 2D Systems Theory, Part II: 2D State Space ModelsRealization and the Notions of Controllability, Observability, and Minimality, Proceedings of the IEEE, Vol.65, No.6, June 1977, pp.945961. [19] S.K. Mitra, A.D. Sagar, and N.A. Pendergrass, Realizations of twodimensional recursive digital filters, IEEE Transactions on Circuits and Systems, CAS22, No.3, March 1975, pp.177184. [20] P.N. Paraskevopoulos, G.E. Antoniou, and S.J. Varoufakis, Minimal state space realization of 3d systems, Proceedings of the IEEE, Vol. 35, No.2, April 1988, pp.6570. [21] G.S. Rao, P. Karivaratharajan, and K.P. Rajappan, On realization of two dimensional digitalfilter structures, IEEE Transactions on Circuits and Systems, CAS23, No.7, 1976, pp.479494. [22] M. Rosenblum, A corona theorem for countable many functions, Integral Equa tions and Operator Theory 3, 1980, pp.125137. [23] W. Rudin, Real and Complex Analysis, McGrawHill, New York 1966. [24] B. Sz.Nagy and C. Foias, Harmonic analysis of operators on a Hilbert space, NorthHolland Publishing Co., Amsterdam, 1970. [25] S.J. Varoufakis, P.N. Paraskevopoulos, and G.E. Antoniou, On the minimal statespace realizations of allpole and allzero 2d systems, IEEE Transactions on Circuits and Systems, CAS34, No.3, March 1987, pp.289292. 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. LiChien 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 ___________ Dean, Graduate School ri0 ILD UNIVERSITY OF FLORIDA IIII IIIIII111 i 11111111111111i I IIIIII11 3 1262 08555 1819 