Workshop on Operator Theory and Complex Analysis: Sapporo, Japan, June 1991 (Operator Theory: Advances and Applications)

OTS9 Operator Theory: Advances and Applications Vol. S9 Editor: I. Gobberg Tel Aviv University Ramat Aviv, Israel Editorial Office: School of Mathematical Sciences Tel Aviv University Ramat Aviv, Israel Editorial Board: A. Atzmon (Tel Aviv) J. A. Ball (Blacksburg) L. de Branges (West Lafayette) K. Clancey (Athens, USA) L. A. Coburn (Buffalo) R. G. Douglas (Stony Brook) H. Dym (Rehovot) A. Dynin (Columbus) P. A. Fillmore (Halifax) C. Foias (Bloomington) P. A. Fuhrmann (Beer Sheva) S. Goldberg (College Park) B. Gramsch (Mainz) J. A. Helton (La Jolla)

M. A. Kaashoek (Amsterdam) T. Kailath (Stanford) H. G. Kaper (Argonne) S. T. Kuroda (Tokyo) P. Lancaster (Calgary) L. E. Lerer (Haifa) E. Meister (Darmstadt) B. Mityagin (Columbus) J. D. Pincus (Stony Brook) M. Rosenblum (Charlottesville J. Rovnyak (Charlottesville) D. E. Sarason (Berkeley) H. Widom (Santa Cruz) D. Xia (Nashville)

Honorary and Advisory Editorial Board: P. R. Halmos (Santa Clara) T. Kato (Berkeley) P. D. Lax (New York)

Birkhauser Verlag Basel . Boston . Berlin

M. S. Livsic (Beer Sheva) R. Phillips (Stanford) B. Sz.-Nagy (Szeged)

Operator Theory and Complex Analysis Workshop on Operator Theory and Complex Analysis Sapporo (Japan) Junel991 Edited by T.Ando I. Gohberg

Birkhiuser Verlag Basel· Boston· BerUn

Editors' addresses:

Prof. T. Ando Research Institute for Electronic Science Hokkaido University Sapporo 060 Japan Prof. I. Gohberg Raymond and Beverly Sackler Faculty of Exact Sciences School of Mathematical Sciences Tel Aviv University 69978 Tel Aviv, Israel

A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Deutsche Bib60thek Cataloging-in-Publication Data Operator theory ad complex ualysis / Workshop on Operator Theory aild Complex Analysis, Sapporo (Japan), June 1991. Ed. by T. Ando ; I. Gohberg. - Basel ; Boston ; Berlin : Birkhiiuser, 1992 (Operator theory; Vol. 59) ISBN 3-7643-2824-X (Basel ... ) ISBN 0-8176-2824-X (Boston ... ) NE: Ando, 1Suyoshi [Hrsg.]; Workshop on Operator Theory and Complex Analysis ; GT

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law, where copies are made for other than private use a fee is payable to »Verwertungsgesellschaft Wort«, Munich.

© 1992 Birkhauser Verlag Basel, P.O. Box 133, CH-4010 Basel Printed in Germany on acid-free paper, directly from the authors' camera-ready manuscripts ISBN 3-7643-2824-X ISBN (1.8176-2824-X

v

Table of Contents Editorial Introduction

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

v. Adamyan Scattering matrices for microschemes . . . . . 1. General expressions for the scattering matrix . 2. Continuity condition References . . . . . . . . . . . . . . . . .

1

2 7 10

D. Alpay, A. Dijksma, J. van der Ploeg, H.S. V. de Snoo Holomorphic operators between Krein spaces and the number of squares of associated kernels . . . . . . O. Introduction . . . . . . . . . . . . . 1. Realizations of a class of Schur functions 2. Positive squares and injectivity . . . . . 3. Application of the Potapov-Ginzburg transform References . . . . . . . . . . . . . . . . . . D. Alpay, H. Dym On reproducing kernel spaces, the Schur algorithm, and interpolation in a general class of domains 1. Introduction . 2. Preliminaries. . . . . . . . . . . . . . . 3. 8(X) spaces . . . . . . . . . . . . . . . 4. Recursive extractions and the Schur algorithm 5. 'Hp( S) spaces . . . . . . . . 6. Linear fractional transformations 7. One sided interpolation 8. References . . . . . . . . . . M. Bakonyi, H.J. Woerdeman The central method for positive semi-definite, contractive and strong Parrott type completion problems . 1. Introduction . . . . . . . . . . 2. Positive semi-definite completions 3. Contractive completions . . . . . 4. Linearly constrained contractive completions References . . . . . . . . . . . . . . . .

IX

..

11 11 15 20

23 28

30

31 33

39 47 57 64 67

74

78 78

79 87

89 95

VI

I.A. Ball, M. Rakowski Interpolation by rational matrix functions and stability of feedback systems: The 4-block case Introduction . . . . . . . . . . . . . 1. Preliminaries. . . . . . . . . . . . 2. A homogeneous interpolation problem . 3. Interpolation problem . . . . . . . . 4. Parametrization of solutions . . . . . 5. Interpolation and internally stable feedback systems ~erences . . . . . . . . . . . . . . . . . . .

.96 .96 100 104 109 116 131 140

H. Bart, V.E. Tsekano1Jskii Matricial coupling and equivalence after extension 1. Introduction . . . . . . . 2. Coupling versus equivalence 3. Examples . . . . . . . . 4. Special classes of operators ~erences . . . . . . . . .

143 143 145 148 153 157

1.1. Fujii Operator means and the relative operator entropy 1. Introduction . . . . . . . . . . . . . . . . . 2. Origins of operator means . . . . . . . . . . . 3. Operator means and operator monotone functions 4. Operator concave functions and Jensen's inequality 5. Relative operator entropy ~erences . . . . . . . . . . . . . . . . . . .

161 161 162 163 165 167

171

M. Fujii, T. Furuta, E. Kamei An application of Furuta's inequality to Ando's theorem 1. Introduction . . . . . . 2. Operator functions . . . . . . . 3. Furuta's type inequalities . . . . 4. An application to Ando's theorem ~erences . . . . . . . . . . . .

173

173 175 176 177 179

T. Furuta Applications of order preserving operator inequalities O. Introduction . . . . . . . . . . . . . . . . . 1. Application to the relative operator entropy . . . 2. Application to some extended result of Ando's one ~erences . . . . . . . . . . . . . . . . . . .

180 180 181 185 190

VII

1. Gohberg, M.A. Kaashoek The band extension of the real line as a limit of discrete band extensions, I. The main limit theorem O. Introduction . . . . . . . . I. Preliminaries and preparations II. Band extensions . . . . . III. Continuous versus discrete References . . . . . . . . .

191 191 193 201 205 219

K. ]zuchi Interpolating sequences in the maximal ideal space of HOC II . . 1. Introduction . 2. Condition (A 2 ) 3. Condition (A3) 4. Condition (Ad References . . .

221 221 223 227 231 232

C.R. Johnson, M. Lundquist Operator matrices with chordal inverse patterns 1. Introduction . 2. Entry formulae 3. Inertia formula References . . .

234 234 237 243 251

P. Jonas, H. Langer, B. TextoriulJ Models and unitary equivalence of cyclic selfadjoint operators in Pontrjagin spaces Introduction . . . . . . . . . . . . . . . 1. The class :F of linear functionals . . . . . 2. The Pontrjagin space associated with ¢> E :F 3. Models for cyclic selfadjoint operators in Pontrjagin spaces 4. Unitary equivalence of cyclic selfadjoint operators in Pontrjagin spaces References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

252 252 253 257 266 275 283

T. Okayasu The von Neumann inequality and dilation theorems for contractions 1. The von Neumann inequality and strong unitary dilation 2. Canonical representation of completely contractive maps 3. An effect of generation of nuclear algebras References . . . . . . . . . . . . . . . . . . .

285 285 287 289 290

L.A. Sakhno'IJich Interpolation problems, inverse spectral problems and nonlinear equations References . . . . . . . . . . . . . . . . . . .

292 303

vm S. Takahashi Extended interpolation problem in finitely connected domains Introduction . . . . . . . . . . . . I. Matrices and transformation formulas II. Disc Cases . . . . . . . . . III. Domains of finite connectivity. R.eferences . . . . . . . . . . . E.B. TsekanotJskii Accretive extensions and problems on the Stieltjes operator-valued functions relations . . . . . . . . . . . . 1. Accretive and sectorial extensions of the positive operators, operators of the class C(9) and their parametric representation 2. Stieltjes operator-valued functions and their realization . 3. M.S. Livsic triangular model of the M-accretive extensions (with real spectrum) of the positive operators . . . . . 4. Canonical and generalized resolvents of QSC-extensions of Hermitian contractions R.eferences . . . . . . . . . . . . . . . . . . . . . V. VinnikotJ Commuting nonselfadjoint operators and algebraic curves 1. Commuting nonselfadjoint operators and the discriminant curve 2. Determinantal representations of real plane curves 3. Commutative operator colligations . . . . . . . . . . . 4. Construction of triangular models: Finite-dimensional case 5. Construction of triangular models: General case . . . 6. Characteristic functions and the factorization theorem R.eferences . . . . . . . . . . . . P.y. Wu AD (?) about quasinormal operators 1. Introduction . . . . . . 2. Representations 3. Spectrum and multiplicity 4. Special classes . . 5. Invariant subspaces 6. Commutant . 7. Similarity . . . . 8. Quasisimilarity . . 9. Compact perturbation 10. Open problems R.eferences . . . .

Workshop Program . List of Participants

305 305 306 309 318 326

328 329 335 345 343 344

348 348 350 353 355 359 364 370 372 372 374 377

379 380 382 385 387 391 393 394

399

402

IX

EDITORIAL INTRODUCTION

This volume contains the proceedings of the Workshop on Operator Theory and Complex Analysis which was held at the Hokkaido University, Sapporo, Japan, June 11 to 14, 1991. This workshop preceeded the International Symposium on the Mathematical Theory of Networks and Systems (Kobe, Japan, June 17 to 21, 1991). It was the sixth workshop of this kind, and the first to be held in Asia. Following is a list of the five preceeding workshops with references to their proceedings: 1981 Operator Theory (Santa Monica, California, USA) 1983 Applications of Linear Operator Theory to Systems and Networks (Rehovot, Israel), OT 12 1985 Operator Theory and its Applications (Amsterdam, the Netherlands), OT 19 1987 Operator Theory and Functional Analysis (Mesa, Arizona, USA), OT 35 1989 Matrix and Operator Theory (Rotterdam, the Netherlands), OT 50 The next Workshop in this series will be on Operator Theory and Boundary Eigenvalue Problems. It will be held at the Technical University, Vienna, Austria, July 27 to 30, 1993. The aim of the 1991 workshop was to review recent advances in operator theory and complex analysis and their interplay in applications to mathematical system theory and control theory. The workshop had a main topic: extension and interpolation problems for matrix and operator valued functions. This topic appeared in complex analysis at the beginning of this century and now is maturing in operator theory with important applications in the theory of systems and control. Other topics discussed at the workshop were operator inequalities and operator means, matrix completion problems, operators in spaces with indefinite scalar product and nonselfadjoint operators, scattering and inverse spectral problems. This Workshop on Operator Theory and Complex Analysis was made possible through the generous financial support of the Ministry of Education of Japan, and also of the International Information Science Foundation, the Kajima Foundation and the Japan Asso-

x ciation of Mathematical Sciences. The organizing committee of the Mathematical Theory of Networks and Systems (MTNS) has rendered financial help for some participants of this workshop to attend MTNS also. The Research Institute for Electronic Science, Hokkaido University, provided most valuable administration assistance. All of this support is acknowledged with gratitude.

T. Ando,

I. Gohberg

August 2, 1992

Operator Theory: Advances and Applications, Vol. 59 © 1992 Birkhauser Verlag Basel

1

SCATTERlNG MATRlCES FOR MICROSCHEMES

Vadim Adamyan A mathematical model for a simple microscheme is constructed on the basis of the scattering theory for a pair of different self-adjoint extensions of the same symmetric ordinary differential operator on a one-dimensional manifold, which consist of a finite number of semiinfinite straight outer lines attached to a "black box" in a form of a flat connected graph. An explicit expression for the scattering is given under a continuity condition at the graph vertices.

Contemporary technologies provide for the formation of electronic microschemes composed of atomic clusters and conducting quasionedimensional tracks as "wires" on crystal surfaces. Electrons travelling along such "wires" behave like waves and not like particles. The explicit expressions for the functional characteristics of simple microschemes can be obtained using the results of the mathematical scattering theory adapted for the special case when different self-adjoint extensions of the same symmetric ordinary differential operator on a one-dimensional manifold 0 are compared. The manifold 0 simulating the visual structure of a microscheme consists of m « 00) semiinfinite straight outer lines attached to a "black box" in a form of a flat connected graph. Different self-adjoint extensions of the mentioned symmetric operator distinguish only in boundary conditions at terminal points of outer lines. Having the scattering matrix for the pair of any self-adjoint operators of such kind with the special extension corresponding to the case of disjoint isolated outer lines one can immediately calculate by known formulae the high-frequency conductance of the "black box" . In the first section of this paper the general expression for the scattering matrix for two self-adjoint extensions of the second-order differential operator in L2(0) is derived on the basis of more general results obtained in [I]. The detailed version of these results is given in the second part for the most important case when functions of the extension domain satisfy the condition of continuity

Adamyan

2

at the points of connection of outer lines to vertices of the graph. This work was initiated by some ideas from the paper [2]. 1. General expressions for the scattering matrix.

Remind the known concepts of the scattering theory. Let L be a Hilbert space and Ho, H be a pair of self-adjoint operators on L. Denote by Po the orthogonal projection on the absolutely continuous subspace of Ho. If the resolvent difference

(H - zI)-l - (Ho - zI)-l at some (and, consequently, at any) nonreal point z is a nuclear operator, then by the Rosenblum-Kato theorem the partial isometric wave operators

W±(H, Ho) = s-l±im eiHte-iHot Po 1-+

00

exist and map the absolutely continuous subspace of Ho onto the same subspace of H [31. The scattering operator

is unitary on PoL and commutes with Ho. The scattering matrix SP), -00 < ,\ < 00, is the scattering operator S(H, Ho) in the spectral representation of Ho. Consider the special case when Hand Ho are different self-adjoint extensions of the same densely defined symmetric operator A in L with finite defect numbers (m, m). Let (ev)r be any basis of the defect subspace ker[A· + ill. Put

ev(z)

= (Ho + il)(Ho -

zI)-lev,

II

= 1, ... , m,

and introduce the matrix-function .1.(z),

.1."v(z)

= (Hoz + I)(Ho -

zI)-le v , e,,),

II,

p.

= 1, ... , m.

According to the M. G. Krein resolvent formula for fixed Ho and arbitrary H m

(1)

(H - zI)-l = (Ho - zI)-l -

L

([.1.(z)

+ QI- l ) "V (., ev("z»)e,,(z)

",v=l

where a parameter Q is a Hermitian matrix [4]. The following parametric representation of the scattering matrix S('\) for extensions H and Ho was derived using the formula (1) in [1].

3

Adamyan

Take any decomposition of the absolutely continuous part Lo of H0 into a direct integral

Lo = [ : ffiK(A) dA. Without loss of generality one can assume dimK(A) :::; m. Let h,,(A) be a spectral image of the vector Poe". Then

(2)

SeA) = 1+ 21ri(A2

+ 1) ~)[~ *(A + iO) + Qt1) 1''' (- , h,,(A» K(>.)hl' (A). 1'."

Now we are going to adapt (2) for the case when a given symmetric operator A is a second-order differential operator on the above mentioned one-dimensional manifold o of the graph Oint and m outer semiaxes.

All self-adjoint extensions of A in L2(0) differ only by boundary conditions at the terminals (e~)i of outer lines and the corresponding connecting points (et)i from the inside of the "black box" Oint. Take as Ho a special extension decomposing into an orthogonal sum : m

Ho = HPnt ffi

L ffiH2j 1:=1

HPnt : L 2 (Oint) H2 : L2(E+)

--+ L2(Oine)j --+

L2(E+), f'(e2) =

(H2f)(z) = !..r(z), I'

o.

As a consequence of the decomposition of Ho the Green function G~(z, y), z, YEO, of the operator Ho, i.e. the kernel of the resolvent [Ho - wlj-1 in L2(0) possesses the property: for any regular point w G",(z, y) = 0 if z E Oint and y belongs to any outer line and vice versa or if z and y are points of different outer lines. Describe the assumed properties of Hfnt. This operator can be considered as a self-adjoint extension of the orthogonal sum A int = E" ffiAi" of regular symmetric differential operators of the form

d

d

Ai" = - deP" (e) de

+ q,,(e)

Adamyan

4

on the corresponding segments (ribs) (a",~,,) of the graph with continuous real-valued functions p,,(e), q,,(e) and p,,(e) > o. The functions f(e) from the Ai" domain in L2(a",~,,) are continuously differentiable and satisfy boundary conditions

f(a,,)

= f(~,,) = OJ

I'(a,,)

= f'(~,,) = O.

Note that every Ai" has defect indices (2N, 2N) where N is the number of the graph ribs. Let B" be the "soft" self-adjoint extension of Ai", i.e. the restriction of the conjugate operator Ai" on the subset of functions f(e) such that

Consider the special self-adjoint extension B = E" €BB" of A int . The operator B can be taken as the part HPnt in the decomposition of Ho. In this case the Green function ~(z, y) on Oint X Oint coincides with the Green function ~(z, y) of Band in its turn E~(z, y) is nonzero only if "z" and "y" belong the same segments (a",~,,) of Oint and on these segments E~(z, y) coincides with the Green functions of B". Note that ~( . ,y) E L2(0) for any regular point wand any yEO. From the definition of the Green function ~(z, y) and its given properties it follows that the functions (G~i(Z, e~»~ together with the functions (G~i(Z, a,,), G~i(Z,~,,»~ form a basis in the defect subspace M == ker[A* + ill of A. Put '12,,-1

= a",

'12"

= ~'"

1/

= 1, ... , Nj

'12N+k

= e~,

k

= 1, ... , mj

and introduce the matrix-function r(w),

According to the Hilbert identity for any regular points w, z of Ho and any z, yEO

(3)

y) - ~(z, y)]. 10.r du ~(z, u)G~(u, y) = _1_[G~(z, w - z

As a consequence of (3) and the relation G~(z, y) = G&(z, y) we have

(4)

~,.,,(w) = =

In ([1 + wHo][Ho - wl]-1~i(·' '1,,»)(Z)~i(Z, r~,,(w) - ilr,.,,(-i) + r",.(-i»).

'1,.)dz

Adamyan

5

Let H be an arbitrary self-adjoint extension of A in £2(0). The Krein resolvent formula (1) and (4) yield the following expression for the Green function Gw(z, y) of H through ~(z,y):

2N+m

Gw(z,y) = ~(z,y) -

(5)

L

([rO(w) +Qtl)/",G~(Z,1/I')G~(1/",Y),

1'.,,=1

where Q is a Hermitian matrix. Now to construct the scattering matrix S('\) for the pair H, H o notice that the parts B" of H o as regular self-adjoint differential operators have discrete spectra and the parts H2 on the outer lines form the absolutely continuous component of Ho. Consequently the first 2N basis vectors G~i(Z, 1/,,) of the defect subspace M_ are orthogonal to the absolutely continuous subspace of Ho and unlike the last m basis vectors G~i(z, T/2N+k) = ~i(Z, e2) belong to this subspace. The natural spectral representation of the absolutely continuous part of Ho, i.e. of the orthogonal sum of the operator H2, is the multiplication operator on the independent variable A in the space of em-valued functions £2(0,00; em). The corresponding spectral mapping of the initial space can be defined in such way that the defect vectors G~i (z, e2) turn into the vector-functions

(6) where

A> 0, ek

E em are the columns

Using all above reasons we get immediately from (2) that S(A) is the (m x m)-matrixfunction and

Represent the parameter Q and the matrix-function rO(w) in the block-diagonal form

(8)

Q = [;.

::],

rO(w) = [r?not(W)

0

-i~lm

]

'

where W is a Hermitian (m x m)-matrix, 1m is the unit matrix of order m and the matrixfunction r?nt(w) is determined by the Green function ~ of the extension B as follows

~.(9)

Adamyan

6

Using (7) and (8) we get the formula:

(10)

S(A) = {i.[¥In

+W

- M*[L

+ r?nt(A -

iO)t 1 M } x

x {-i.[¥In + W - M*[L + r?nt(A _ iO)t 1 M}-1 This expression describes all possible scattering matrices for microschemes comprised of given m outer lines and a given set of N ribs. The parametric matrices W, M,

L contain information on the geometrical structure of the "black box" and the boundary conditions at all vertices including the connecting points to the outer lines. Without loss of generality we can consider that all connecting points are the graph vertices. Single out now the scattering matrices for microschemes which differ only by the way the outer lines are connected to the definite inputs of the "black box", i.e. to the certain vertices of the definite graph

n.

Notice that this limiting condition generally

speaking leaves the Hermitian matrix W arbitrary. The matrix can now vary only as far as the subspace ker M M* remains unchanged. In what follows we will assume that this subspace and, respectively, its orthogonal complement in

[:2N

are always invariant

subspaces of the L matrix. Let Co be the orthogonal projector on the subspace ker M M* in

[:2N.

Under the above condition and for various types of connection of outer lines to

certain graph vertices only the block CoLCo of the L matrix is modified. Notice that this block for a "correct" connection is always invertible. Consider now the connections for which the L matrix in (8) remains unchanged. This matrix is a parameter in the Krein formula when resolvents of B and a definite self-adjoint extension HPnt of Aint are compared. Let rint(w) be the matrix-function which is determined by the Green function E", of Hf:.t like r?nt(w) in (9) by E~. According to the Krein formula of the form (5) (11)

rint(W) = r?nt(w) - r?nt (w)[r?nt (w) = L[r?nt(w)

+ Lt 1r?nt(w)

+ Lt 1r?nt(w) =

L - L[r?nt(w)

+ Lt 1L.

Taking into account that by the assumption that the block CoLCo of the Hermitian matrix L is invertible on the subspace ker M M* C [:2N, denoting by Q the corresponding inverse operator in this subspace and using (11) we can write

(12)

M*[r?nt(w)

+ L1- 1 M = M*QL[r?nt(w) + L1- 1 LQM = M*QM - M*Qrint(w)QM.

Adamyan

7

Inserting the last expression into (10) we find that the scattering matrix S(A) for the connections without changing the parameter L and the subspace ker MM· has a form

(13)

where the matrix parameters

W(= W - M·QM),

M = QM

depend on the boundary conditions at those vertices of the graph 0, which are connected with the outer lines. If there are reasons to consider that as a result of the connection of outer lines to the graph 0 the matrices Land M are changed into M', L' so that ker(L - L') = ker M'· M' = ker M· M, then it is natural to use the representation (14)

S(A) = [iJ¥"In + H(A - iO)] [-iJ¥"In + H(A _ iO)] -1, H(w) = W + M'· [L + r'nt(w)(PO - QL' Po)] -1 x [por.nt(w)Qo - I] M'.

The formula (14) can be obtained from (10) using the relation (11).

2. Continuity condition. From the physical point of view the most natural are self-adjoint extensions of

A satisfying the continuity condition at the graph vertices. This condition states that all functions from the domain of any such extension possess coinciding limiting values at any vertex along all ribs incident to this vertex. Irrespective to the present problem consider now the structure of the Krein formula (5) with parameter Q for arbitrary self-adjoint extensions satisfying the continuity condition in every vertex. Take a vertex with 8 incident ribs, i.e. the vertex of degree 8. It is convenient to enumerate the extreme points of the ribs at the vertex as TJl, .. . , TJ.. Replacing x in Eq. (5) by TJ. for arbitrary y we obtain (15)

O",(TJ., y) =

L Q.,.([rO(w) + Q]-I) ,,"~(TJ", y). ,.,11

It follows from the continuity condition that

(16)

0",('110 y)

= 0",(1/2, y) = ... = 0",('1., y).

Adamyan

8

Since y and w in (15) and (16) are arbitrary it is obvious that the matrix Q in fact transforms any vector from C 2N +m into a vector with equal first 8 components. Denote by J, the matrix of order 8 all components of which are unity. As Q is Hermitian, it is nothing but the following block matrix

where h is a real constant and Q' is a Hermitian matrix of the rank 2N + m -

8.

Since the

same procedure is valid for any vertex of arbitrary degree, the matrix with the suitable enumeration of the extreme points of the ribs and outer lines takes the block-diagonal form

o o. 1 ,

(17)

o where 1 is the total number of the graph vertices and of the vertices. Thus the following lemma is valid.

h,l., 81, ... ,81

are corresponding degrees

LEMMA. The parameter Q in the Krein formula (5) for extensions satisfying

the continuity condition is the Hermitian matrix such that nonzero elements of every its row (column) are equal and situated at the very places where the unities of the incidence matrix of the graph 0 are. Let the matrix Q be already reduced to the block-diagonal form (17) by a corresponding enumeration of extreme point of ribs and terminal points of outer lines.

In this case the matrices W, M and CoLCo of the representation (8) coincide with the diagonal matrix h1 ID>= [

o where the parameters h 1 , • •. ,hm are determined by the boundary conditions in vertices to which the outer lines are connected. Using this fact and (13) we infer: THEOREM. Let H be a self-adjoint extension of A in L2(0) satisfying the continuity condition and univalently connected with the set of parameters h 1 , ••• , h, of the corresponding matrix Q of the form (17) generating H in accordance with the Krein formula and let Ho be the special extension of A decomposing into an orthogonal sum of

Adamyan

9

the self-adjoint operators on the graph and on the outer lines. The scattering matrix S(A) for the pair Ho, H admits the representation (18)

where f.k(W) = E",(f/., f/k) and E",(e, f/) is the Green function of the self-adjoint extension HPnt of A. nt satisfying the continuity condition and determined by the same set of parameters hI, ... , h, for the same vertices like H is. From the representation (18) it is obvious that the analytic properties of the scattering matrix S(A) are essentially determined by those of the matrix f(w) constructed by the Green function of the separated graph. For the regular differential operator the matrix (E",(f/., f/k))~ is the meromorphic R-function. The natural problem thus arises of the partial recovery of the graph structure and the operator on it from the matrix S(A) or, equivalently, by the matrix f. In the case when the graph is reduced to a single segment this problem is the well-known problem of recovery of a regular Sturm-Liouville operator from spectra of two boundary problems. We hope to carry out the consideration of the former problem in a more general case elsewhere. In conclusion, as an example, consider an arbitrary graph with only two outer lines connected to the same vertex. In this case S(A) is the second order matrix-function but the determining matrix f(w) is degenerate and takes the form

where

eis the internal coordinate of the vertex of the graph tangent to the outer lines. The scattering matrix according to (18) now can be put in the usual form

where

i r(A) - - - - - = = - - 2E>.(e,eh!A/1' - i' are, respectively, the reflection and transition coefficients. Notice that according to the Landauer formula the resistance of the graph is given by

where Ro is the quantal resistance, i.e. the universal constant.

Adamyan

10

REFERENCES 1.

Adamyan, V. M.j Pavlov, B. S.: Null-range potentials and M. G. Krein's formula of generalized resolvents (in Russian), Studies on linear operators offunctions. xv. Research notes ofscientific seminars of the LBMI, 1986, v.149, pp. 723.

2.

Exner, P.; Seba, P.: A new type of quantum interference transistor, Phys. Lett. A 129:8,9 (1988), 477-480.

3.

Reed, M.; Simon, B.: Methods of modern mathematical physics. III: Scattering theory, Academic Press, New York - San Francisco - London, 1979.

4.

Krein, M. G.: On the resolvents of a Hermitian operator with the defect indices (m, m) (in Russian), Dokl. Acad. Nauk SSSR 52:8 (1946), 657-660.

Department of Theoretical Physics University of Odessa, 270100 Odessa Ukraine MSC 1991: 8tU,47A40

11

Operator Theory: Advances and Applications, Vol. 59 © 1992 Birkhliuser Verlag Basel

UOWMORPIDC OPERATORS BETWEEN KREIN SPACES AND THE NUMBER OF SQUARES OF ASSOCIATED KERNElS

D. Alpay, A. Dijksma, J. van der Ploeg, U.S.V. de Snoo Suppose that e(z) is a bounded linear mapping from the Krein space 15 to the KreIn space ~, which is defined and holomorphic in a small neighborhood of z = O. Then often e admits realizations as the characteristic function of an isometric, a coisometric and of a unitary colligation in which for each case the state space is a KreIn space. If the colligations satisfy minimality conditions (i.e., are controllable, observable or closely connected, respectively) then the positive and negative indices of the state space can be expressed in terms of the number of positive and negative squares of certain kernels associated with e, depending on the kind of colligation. In this note we study the relations between the numbers of positive and negative squares of these kernels. Using the Potapov-Ginzburg transform we give a reduction to the case where the spaces 15 and ~ are Hilbert spaces. For this case these relations has been considered in detail in [DiS!]. O.

INTRODUCTION

Let (0', [.,.]1\') and

(~,

["']l!!), or 15 and ~ for short, be Krein spaces and denote by L(15,~)

the space of bounded linear operators from 15 to write

r

~

(we write L(O') for L(lJ,15)). If

TEL(lJ,~),

we

(EL(~,!J)) for the adjoint of T with respect to the indefinite inner products [.,.]1\' and

["']l!! on the spaces !J and~.

invertible) if y-l EL(~,!J,). valued functions

We say that

TEL(lJ,~)

is invertible (instead of boundedly

By S(O',~) we denote the (generalized) Schur class of all L(lJ,~)

e, defined and holomorphic on some set in the open unit disc

D={ZEC Ilzl < I};

we denote by :b(e) the domain of holomorphy of e in D. The class of Schur functions e for which OE:b(e) will be indicated by S°(O',~).

If:b is a subset of D, we write :b*={zlzE:b}.

eES(!J,~) we associate the function e defined by e(z) = e(z)*.

and if eES°(lJ,~), then eES°(lJ,~). We associate with O'e(Z,W) = I-e(w) *e(Z), l-wz

with values in L(!J) and

L(~),

z,wE:b(e),

e the

O'e(Z,W)

respectively, and the kernel

With each

Clearly, eES(~,15), :b(e) = :b(e)*

kernels

l-e(w)e(z)* l-wz

z,WE:b(e)*,

12

Alpayetal.

Sa(Z,'ID) = [

I-8~:~:8(Z)

8(Z):=:('ID)* ),

8(w)-8(z)

w-z with values in

L(lJGl~),

where

z,'IDe:b(8)n:b(8)*,

z~w,

I-8(iO)8(z)*

l-wz O'Gl~

stands for the orthogonal sum of the Krein spaces 0' and

~.

Here I is the generic notation for the identity operator on a Krein space, so in the kernels I on~.

is the identity operator on 0' or

If we want to be more specific we write, e.g., Ifj to

indicate that I is the identity operator on 0'.

In this paper we prove theorems about the

relation between the number of positive (negative) squares of the matrix kernel Sa and those of the kernels ua and ua on the diagonal of Sa. We recall the following definitions.

Let st be a Krein space.

A kernel K(z,'ID) defined for Z,'ID

in some subset :b of the complex plane ( with values in L(st), like the kernels considered above, is called nonpositive (nonnegative), if K(z,'ID)* = K('ID,z), z,'IDe:b, so that all matrices of the form ([K(zi,zj)fi,filltl7. i =1> where neN,

zl> ... ,zne:b and fl> ... ,fnest are arbitrary, are hermitian,

and the eigenvalues of each of these matrices are nonpositive (nonnegative, respectively).

More

generally, we say that K(z,'ID) has /\, positive (negative) squares if K(z,'ID)*=K('ID,z), z,'IDe:b, and all hermitian matrices of the form mentioned above have at most/\' and at least one has exactly/\, positive (negative) eigenvalues.

It has infinitely many positive (negative) squares if for each

/\, at least one of these matrices has not less than /\, positive (negative) eigenvalues.

We denote

the number of positive and negative squares of K by sq+(K) and sqJK), respectively.

If, for

example, sq_(K) =0, then K(z,'ID) is nonnegative. In the sequel we denote by ind+st and ind_st the dimensions of the Hilbert space st+ and the anti Hilbert space st_ in a fundamental decomposition .It=st+Glst_ of st. Then ind±st=sq±(K) where K is the constant kernel K(z,'ID) =1, and the indices are independent of the chosen fundamental decomposition of st.

Whenever in this paper we use the

term Pontryagin space we mean a Krein space, st, say, for which ind_st < 00. The main theorems in this paper concern the relation between the values of sq±(Sa) on the one hand and the values of sq±(ua) and sq±(ue) on the other hand.

The most general one implies

that, if /\'eNu{O}, then sq_(Sa)=/\' if and only if sq_(Ua) =sq_(ua) =/\" and sq+(Sa)=/\' if and only if sq+(ua)=sq+(ue)=/\,. To formulate this theorem we consider two fundamental decompositions lJ'=lJ'+GllJ'_ and ~=~+Gl~_ of 0' and~, and we denote by P± the orthogonal projections on 0' onto the spaces O'± and by Q± the orthogonal projections on ~ onto the spaces ~±. THEOREM

0.1. Let 0' and ~ be Krein spaces and let 8eS(0',~).

Then:

(i) sq_(Sa)gze(g we have if Ll is isometric;

(i)

[u H (z,w)fl>fz)!j = [(l-zTfIFfl>(l-wT)-IFf2)lt,

(ii)

[ue(z,w)gl,gz)l!I = [(l-zT*flC*gl,(l-wT*)-IC*gZ)lt,

(iii)

[Sa(z,w) (~:),

if Ll is coisometric;

(~:) )!jel!l= [(I -zTflFfl +(l-zT*fICx'*gl>(l-wTfIFfz+(I-w1'*flG*g2ht. if Ll is unitary.

In this section we summarize some of the results from [DLSl).

17

Alpayet al. THEOREM 1.2.

Assume that

0'

and ® are Hilbert spaces and let eES°(B',®).

Then

That is, either these three numbers are infinite or one of them is finite and then they all are finite and equal.

If one of them is finite and equal to II:EINU{O}, say, then 13 admits the

following realizations. (a) e=eLl,on :b(e) for some isometric colligation 4 1= (.R\,O',®;U1), and then ind±.R:1~sq±(us); the isometric colligation 41 can be chosen closely innerconnected, in which case it is uniquely determined up to isomorphisms and ind±.R:1 = sq±(us). (b) e=e Ll2 on :b(e) for some coisometric colligation 4 2 = (.R: 2,O',®;U 2), and then coisometric colligation 42 can be chosen closely outerconnected,

ind±.R:2~sq±(ue);

the

in which case it is

uniquely determined up to isomorphisms and ind±.R:2 = sq±(ue). (c) e=e Ll on :b(e) for some unitary colligation 4= (.R:,O',®;U), and then

ind±.R:~sq±(Ss);

the unitary

colligation 4 can be chosen closely connected, in which case it is uniquely determined up to isomorphisms and ind±.R: = sq±(Ss). One of the key tools in the proof of Theorem 1.2 given in [DLS1] is formulated in the following lemma.

A Hilbert space version of it is given by de Branges and Shulman in [BS].

It

relates isometric colligations to unitary colligations. LEMMA 1.3.

Let 41 = (.R:1,O',®;U.) be a closely innerconnected isometric colligation, in which

and ® are Hilbert spaces and .R:1 is a Pontryagin space.

0'

Then there exists a closely connected

unitary colligation 4=(.R:,O',®;U) such that ind_.R:=ind_.R: b UIB'=V and eLl=eLl,on :b(eLl)n:b(eLl,). Sketch of the proof of Theorem 1.2.

(1) We first consider the case where sq_(us) is finite,

and we briefly describe the construction of the realization described in (a). linear space £ of finite sums ['zczf., where zE:b(e), fzEO' and

Cz

Consider the

is a symbol associated with

each zE:b(e) and provide £ with the (possibly degenerate, indefinite) inner product [[,. cJ.,[,w cwgw] = ['z ,)us(z,w)f.,gw) ]B" Define the linear operators To, Fo, Go and H via the formulas 1

Toc.f =z(czf-cof), GoE:J =~(e(Z)f -e(O)f),

Hf=e(O)f,

where Z # 0 and fEB'. Then To and Go are densely defined operators on £ with values in £ and ®, respectively, Fo:O' -. £, H:O' ~ ®,

is an isometric operator on a dense set and H+zGo(I-zT of 1Fo=e(z) on

0'.

Now we consider the

18

Alpayet aI.

quotient space of £ over its isotropic part and redefine the operators on this space in the usual way. Then completing the quotient space to a Pontryagin space and extending the operators by continuity to this completion we obtain a closely innerconnected isometric colligation with the desired properties. (2) If it is given that sq_(ue) is finite we apply the above construction to realization in terms of a closely innerconnected isometric colligation .Ill =

e to

obtain a

(st'!l~,8';Utl.

Then

O,

i=I,2, ... ,n.

Then the nxn matrix (Qij/(I-zizj )) is

positive.

Let P be the matrix (Qij/(I-zizj )).

Then P= r:_oOkQDook, where O=diag(zHz2, ... ,zn)' Hence P~O, and to show that P>O, it suffices to prove that R(P)={O}, where R(P)cCn is the null Proof.

space of P. Let u=(ui)eR(P). Then O=u*Pu= r:=o(Oooku)*QDooku, and, since each summand in the series is nonnegative, the vector QD."", = 0, k = 0,1,.... with complex coefficients the vector QP(O*)u=O.

It follows that for every polynomial P

Let ie{I,2, ... ,n} and let Pi be a polynomial

with the property that P i(zj)=6ij , the Kronecker delta. Then O=(QPi(O*)U)i=QiiUi, which implies that Ui=O, since, by assumption, Qii>O. It follows that "'=(Ui)=O, i.e., R(P)={O}. Note that in the proof of Lemma 2.1 we have used the Taylor expansion 1/(I-zw)= r:=dwz)n. A similar proof can be given based on the integral representation of 1/ (1- zw) given in the Introduction just above Theorem 0.2. THEOREM

(i)

2.2. Let 0' and

~

be Hilbert spaces and let 8eS(0',~).

If sq+(O'e) .) = X(>.)JX(w)* w pw(>') , where X is a k x m matrix valued function, J is an m X m signature matrix (i.e., J and J J* = 1m), the denominator Pw(>') is of the form

(1.1)

= J*

Pw(>') = a(>.)a(w}* - b(>.)b(w)* ,

(1.2)

and it is further assumed that: I.

II.

a(>.) and b(>.) are analytic in some open nonempty connected subset 0 of CV. The sets

0+ = {w EO: Pw(w) > O}

0_ = {w EO: PIAI(w) < O}

and

are both nonempty. Because of the presumed connectedness of 0, it follows further (see [AD4] for details) that: III.

The set

00

= {w EO:

= O} o.

PIAI(w)

contains at least one point I" such that pp(>') ~

Any function PIAI(>') of the form (1.2) which satisfies I and II (and hence III) will be said to belong to 'Do. Reproducing kernel Pontryagin spaces with reproducing kernels of the form (1.1) were extensively studied in [AD3] for the special choices PIAI(>') = 1 - >.w* and PIAI(>') = -27ri(>. - w*). Both of these belong to 'Do with 0 = CV:

= 1, b(>.) = >., 0+ = ID and 00 = ']['. w*) is of the form (1.2) with a(>.) = y'i(l- i>.), b(>.) = y'i(l + i>.),

1 - >.w* is of the form (1.2) with a(>.)

-27ri(>. 0+ = CV+ and 00 = JR.

In this paper we shall extend some of the results reported in [AD3] to this new more general framework of kernels with denominators in 'Do and shall also solve a general one-sided interpolation problem in this setting. Many kernels can be expressed in the general form (1.1); examples and references will be furnished throughout the text. In addition to these, which focus on AIAI(>') as a reproducing kernel, the form (1.1) shows up as a bivariate generating function in the study of structured Hermitian matrices; see the recent survey by Lev-Ari [LA] and the references cited therein. In particular, Lev-Ari and Kailath [LAK] seem to have been the first to study "denominators" PIAI(>') of the special form (1.2). They showed that Hermitian matrices with bivariate generating functions of the form (1.1) can be factored efficiently whenever p",(>') is of the special form (1.2). The present analysis gives a geometric interpretation of the algorithm presented in [LAK] in terms of a direct sum orthogonal decomposition of the underlying reproducing Pontryagin spaces.

Alpay and Dym

32

As we already noted in [ADl], the important kernel

Kt./(A) = J - 6(A)J6(w)* Pt./(A)

(1.3)

can also be expressed in the form (1.1), but with respect to the signature matrix

. [J0

J =

0]

-J

'

by choosing X=[Im 6]. For the most part, however, we shall take J equal to

and shall accordingly write

X(A) = [A(A) B(A)] with components A E ([!kxp and B E ([!kxq, both of which are presumed to be meromorphic in n+. Then (1.1) can be reexpressed as

At./(A) = A(A)A(w)* _ B(A)B(w)* , Pt./(A) Pt./(A) which serves to exhibit At./(A) as the difference of two positive kernels on n+ (since 1/p",(A) is a positive kernel on n+, as is shown in the next section). Therefore, by a result of L. Schwartz [Sch], there exists at least one (and possibly many) reproducing kernel Krein space with At./(A) as its reproducing kernel. However, if the kernel is restricted to have only finitely many negative squares (the definition of this and a number of related notions will be provided in Section 2), then there exists a unique reproducing kernel Pontryagin space with At./(A) as its reproducing kernel. This too was established first by Schwartz, and independently, but later, by Sorojonen [So] (and still independently, but even later, by the present authors in [AD3]). If At./(A) has zero negative squares, i.e., if At./(A) is a positive kernel, then the (unique) associated reproducing kernel Pontryagin space is a Hilbert space, and the existence and uniqueness of a reproducing kernel Hilbert space with At./(A) as its reproducing kernel also follows from the earlier work of Aronszajn [Ar]. Throughout this paper we shall let 8(X) [resp. .qe)] denote the unique reproducing kernel Pontryagin (or Hilbert) space associated with a kernel of the form (1.1) [resp. (1.3)]. The reproducing kernel Hilbert spaces 8(X) and K(9), but with p",(A) restricted to be equal to either 1- AW* or -27ri(A -w*), originate in the work of de Branges, partially in collaboration with Rovnyak; see [dBl], [dBR], [dB3], the references cited therein, and also Ball [Bal]. Such reproducing kernel Hilbert spaces were applied to inverse scattering and operator models in [AD 1] and [AD2], to interpolation in [Dl] and

Alpay and Dym

33

[D2], to the study of certain families of matrix orthogonal polynomials in [D3] and [D4], and to the Schur algorithm and factorization in [AD3]; the latter also extends a number of basic structural theorems from the setting of Hilbert spaces to Pontryagin spaces. In [AD4] and [AD5], the theory of K(8) spaces was extended beyond the two special choices of p mentioned above, to the case of general p E Do. The parts of that extension which come into play in the present analysis (as well as some other prerequisites) are reviewed in Section 2. In this paper we shall carry out an analogous extension for the spaces 8(X). This begins in Section 3. Recursive reductions and a Schur type algorithm are presented in Section 4. Section 5 treats the special case in which Aw{A) is positive and of the special form Aw(A) = {Ip - S(A)S(w)*}jPw(A). Section 6 deals with linear fractional transformations, and finally, in Section 7, we apply the theory developed to that point to solve a general one-sided interpolation problem in f!+. The basic strategy for solving the interpolation problem in f!+ is much the same as for the classical choices of the disc or the halfplane except that now we seek interpolants S for which the operator Ms of multiplication by S on an appropriately defined analogue of the vector Hardy space of class 2 is contractive: IIMsll :5 1. Although this implies that S is contractive, the converse is not generally true; see the examples in Section 5. Moreover, f!+ need not be connected. Interpolation problems in nonconnected domains have also been considered by Abrahamse [Ab], but both the methods and results seem to be quite different. Finally, we wish to mention that there appear to be a number of points of contact between the interpolation problem studied in this paper and the interpolation problem described by Nudelman [N] in his lecture at the Sapporo Workshop. However, we cannot make precise comparisons because we have not yet seen a written version. The notation is fairly standard: The symbols rn. and ([; denote the real and complex numbers, respectively; ID = {A E ([; : IAI < 1}, 11' = {A E ([; : IAI = 1}, IE = {A E ([; : IAI > 1} and ([;+ [resp. ([;_] stands for the open upper [resp. lower] half plane. ([;pxq denotes the set of p x q matrices with complex entries and ([;P is short for ([;Px 1. A * will denote the adjoint of a matrix with respect to the standard inner product, and the usual complex conjugate if A is just a number.

2. PRELIMINARIES To begin with, it is perhaps well to recall that a vector space V over the complex numbers which is endowed with an indefinite inner product [ , ] is said to be a Krein space if there exist a pair of subspaces V+ and V_ of V such that

(1)

V+ endowed with [ , ] and V_ endowed with -[ ,

(2)

V+ n V_ = {OJ.

(3)

V+ and V_ are orthogonal with respect to [ ,

I and

I are Hilbert

spaces.

their sum is equal to V.

V is said to be a Pontryagin space if at least one of the spaces V+, V_ is finite dimensional. In this paper, we shall always presume that V_ is finite dimensional. In this instance, the dimension of V_is referred to as the index of V.

Alpayand Dym

34

A Pontryagin space P of m x 1 vector valued meromorphic functions defined on an open nonempty subset 6 of cr:, with common domain of analyticity 6', is said to be a reproducing kernel Pontryagin space if there exists an m x m matrix valued function Lw(.~) on 6' x 6' such that for every choice of W E 6', v E cr: m and I E P;

(1)

Lwv E P, and

(2)

[I, Lwv]p = v* I(w).

The matrix function Lw(,x) is referred to as the reproducing kernel; there is only one such. Moreover, for every choice of a and j3 in 6'. The kernel Lw(,x) (or for that matter any Hermitian kernel) is said to have negative squares in 6' if (1) for any choice of points WI, . .. ,W n in 6' and vectors V}, •• . , Vn in cr: m the n x n matrix with ij entry equal to vi LWj (wdvj has at most II negative eigenvalues and, (2) there is a choice of points WI, . .. ,Wk and vectors VI, • •. ,Vk for which the indicated matrix has exactly II negative eigenvalues; it should perhaps be emphasized here that n is also allowed to vary. In a reproducing kernel Pontryagin space, the number of negative squares of the reproducing kernel is equal to the index of the space. II

For additional information on Krein spaces and Pontryagin spaces, the monographs of Bognar [Bo], lohvidov, Krein and Langer [IKL], and Azizov and lohvidov [AI] are suggested. Next, it is convenient to summarize some facts from [AD4] about the class 'Do which was introduced in Section 1 and on some associated reproducing kernel spaces. First, it is important to note that the definition of the class 'Do depends only upon p and not upon the particular choice of functions a and b in the decomposition (1.2). In particular, if Pw(,x) can also be expressed in terms of a second pair of functions c(,x) and d(A): if

PW(,x) = c(,x)c(w)* - d(,x)d(w)* , then there exists a Jll unitary matrix M such that [c(,x) d(,x)]

= [a(,x)

b(,x)]M

for every ,x E n;see Lemma 5.1 of [AD4]. We have already remarked that the functions Pw(,x) -211"i(A - w*) belong to 'Do. So does the less familiar choice

=1-

,xw* and Pw(,x)

PW(,x) = -271"i(,x - w*)(1 - ,xw*) .

=

(2.1)

The latter is of the form (1.2) with

a(,x)

= Ji"p + i(,x2 + I)}

Moreover, in this case,

and b(,x)

= Ji"p -

i(,x2

+ I)}

.

(2.2)

35

Alpay and Dym

is not connected.

S(A)

Now if p E Do. with decomposition (1.2), then a(A) f:. 0 for A E Q+ and strictly contractive in Q+. Therefore, the kernel

= b(A)/a(A) is

kw(A)

1

1

~

=( ') = ( \ L pwA aA)

1

s(A) s(w)

1=0

*1

1 aw)*

-(-

is positive on Q+: for every positive integer n, and every choice of points Q+ and constants CI,' .. , Cn in a:: ,

WI, ... ,W n

in

n

L

cjkw;(Wj)Ci ~ 0 .

i,j=1

Thus, by one of the theorems alluded to in the introduction, there f'xists a unique reproducing kernel Hilbert space, with reproducing kernel kw(A) = l/(h... (>..). We shall refer to this space as Hp and shall designate its inner product by ( )f1p' Recall that this means that, for every choice of W E Q+ and f E H p, (1) 1/ pw belongs to H p, and (2) (f, 1/ Pw) f1p = f(w). The space Hp plays the same role in the present setting as the classical Hardy spaces H2(ID) for the disc and H2( a::+) for the open upper half plane. Indeed, it is identical to the former [resp. the latter] when Pw(A) = 1- AW* [resp. Pw(A) = -27l"i(A-

w*)]. More generally,

H;:'

will denote the space of m x 1 vector valued functions

f with coordinates /; and 9i, i

9

= 1, ... ,m,

in Hp and inner product m

L(fi,9i)f/p i=1

From now on, we shall indicate the inner product on the left by (f, g) f1p (i.e., we drop the superscript m), in order to keep the notation simple. For allY m x Tn sigllature matrix J, the symbol Hp,J will denote the space H',;' endowed with the indefinite inner product

The space H;:' is a reproducing kernel Hilbert space with reproducing kernel Kw(A) = 1m/ Pw(A), whereas Hp,J is a reproducing kernel Krein space with reproducing kernel Kw(.~) = J/Pw(A).

36

Alpay and Oym

Because 1/ Pw(>.) is jointly analyticfor A and w* in out in more detail in [AD4]) that

n+, it follows (as is spelled

a

k 1 1 'Pw,k = -k'. uW ~ *kPw

belongs to H p for every integer k ;::: 0 and that

for every

I

E H p and every w E

n+.

We shall refer to the sequence 'Pw,O,· .. ,'Pw,n-l

n+.

as an elementary chain of length n based on the point w E

ft, ... , In

More generally, by a chain of length n in of the m X n matrix valued function

H;:'

we shall mean the columns

F("\) = Vq,w,n(..\) , wherein V is a constant m x n matrix with nonzero first column and

!

'PW'O(A)

41",n( A)

~

[

'Pw,n-l (..\)

1 (2.4)

o

'Pw,o(..\)

is the n x n upper triangular Toeplitz based on 'Pw,o(..\), ... , 'Pw,n-l("\) as indicated just above. It is important to note that

(2.5) where Aw and Bw are the n x n upper triangular Toeplitz operators given by the formulas

o

o

r1 *

and

Bw

!l' ,

(2.6)

130

00

where a(j)(w)

OJ = --.,}.

and

13j

b(j)(w) = --.,- . }.

These chains are the proper analogue in the present setting of the chains of rational functions considered in [A03] and [02]. They reduce to the latter in the classical cases, i.e., when Pw(A) = 1 - AW* or Pw(A) = -21ri(A - w*).

37

Alpay and Dym

In the classical cases, every finite dimensional space of vector valued meromorphic functions which is invariant under the resolvent operators (RoJ)()..) = f()..) - f(o) )..-0

(2.7)

(for every a in the common domain of analyticity) is made up of such chains. An analogous fact holds for general p E Vn, but now the invariance is with respect to the pair of operators a()..)f()..) - a(o)f(o) {r( a, b; a )f}()..) = a( a )b()..) _ b( o)a( )..)

(2.8)

b()..)f()..) - b(o)f(o) {r(b,a;o)f}()..) = b(o)a()..) _ a(o)b()..) ,

(2.9)

and

see [AD5] for details. Just as in the classical cases, a non degenerate finite dimensional subspace of Hp,J with a basis made up of chains is a reproducing kernel Pontryagin space with a

reproducing kernel of the form (1.3). More precisely, we have: THEOREM 2.1. Let p E Vn and let A E ([;nxn, B E ([;tlxn and V E ([;mxn be a given set of constant matrices such that

(1)

det{a(",)A - b(",)B} =I- 0 for some point", E no, and

(!)

the columns of

F()")

= V{a()")A -

b()")B}-1

(2.10)

are linearly independent (as analytic vector valued functions of)..) in n~, the domain of analyticity of F in n+. Then, for any invertible n X n Hermitian matrix P, the space

:F = span {columns of F()")} , endowed with the indefinite inner product

(2.11)

is an n dimensional reproducing kernel Pontryagin space with reproducing kernel K..,()..) = F(),,)P- I F(w}* .

The reproducing kernel can be expressed in the form

K ()..) = J - 9()")J8(w)*

'"

p",()..)

(2.12)

Alpay and Dym

38

for some choice of m x m signature matrix J and m x m matrix valued function 6(..\) which is analytic in n~ if and only if P is a solution of the equation A * P A - B* P B = V* JV .

(2.13)

Moreover, in this instance, 6 is uniquely specified by the formula (2.14)

with JJ as in (1), up to a J unitary constant factor on the right. PROOF. This is Theorem 4.1 of [AD4]. An infinite dimensional version of Theorem 2.1 is established in [AD5]. Therein, the matrix identity (2.13) is replaced by an operator identity in terms of the operators r( a, b, a) and r( b, a, a). In the sequel we shall need two other versions of Theorem 2.1: Theorems 5.2 and 5.3 of [AD4], respectively. They focus on the special choice of A = Aw and B = Bw. THEOREM 2.2. Let JJ and F be as in Theorem 2.1, but with A = Aw and B = Bw for some point wE n+. Then the columns II, ... '/n of F belong to H;: and the n x n matrix P with ij entry

is the one and only solution of the matrix equation (2.15)

THEOREM 2.3. Let p E Do., JJ E no and w E n+ be such that p,..(w) =f. 0 and suppose that the n x n Hermitian matrix P is an invertible solution of (2.15) for some m X m signature matrix J and some m X n matrix V of rank m with nonzero first column. Then the columns II, ... ,fn of

F(>.) = V{a(..\)Aw - b(..\)Bw}-1 are linearly independent (as vector valued functions on n~) and the space:F based on the span of II, ... , fn equipped with the indefinite inner product

e

(for every choice of and TJ in a: n ) is a K(6) space. Moreover, 6 is analytic in n+ and is uniquely specified by formula (2.14), up to a constant J unitary factor on the right. the class in 0'+.

From now on we shall say that the m x m matrix valued function 6 belongs to if it is meromorphic in n+ and the kernel (1.3) has v negative squares

'P~(n+)

Alpay and Oym

39

3. 8(X) SPACES is an m

X

Throughout this section we shall continue to assume that p E Vn and that J m signature matrix.

A k x m matrix valued function X will be termed (S1+, J,p)v admissible if it is meromorphic in S1+ and the kernel

Aw(-\) = X(-\)JX(w)* Pw(,\)

(3.1)

has v negative squares for -\ and w in S1~, the domain of analyticity of X in S1+. Every such (S1+, J, p)v admissible X generates a unique reproducing kernel Pontryagin space of index v with reproducing kernel Aw(-\) given by (3.1). When v = 0, X will be termed (S1+, J, p) admissible. In this case, the corresponding reproducing kernel Pontryagin space is·a Hilbert space. We shall refer to this space as SeX) and shall discuss some of its important properties on general grounds, in the first subsection, which is devoted to preliminaries. A second description of SeX) spaces in terms of operator ranges is presented in the second and final subsection. The spaces SeX) will play an important role in the study of the reproducing kernel space structure underlying the Schur algorithm which is carried out in the next section. It is interesting to note that the kernel K S -\

_

w( ) -

Ip - S(-\)S(w)* 1 - -\w· [ S(-\*)* _ S(w)* -\ - w"

S(-\) - S~w*) -\-w Iq - S(-\*)* S(w*) 1 - -\w"

1 ,

based on the p x q matrix valued Schur function S can be expressed in the form (3.1) by choosing -\S(-\) S(-\) ] Ip X(-\) = y'2; 271' [ Up ,\l , S(,\ *)* -\S(,\ *)* Iq q

~

-~ql

0 0 J (-i) 0 Iq and Pw(-\) as in (2.1). This kernel occurs extensively in the theory of operator models; see e.g., [Ba2j, [OLSj and [dBSj.

[-i'

Ip 0 0 0

3.1. Preliminaries. THEOREM 3.1. If the k x k matrix valued function Aw(-\) defined by (9.1) ha", v negative ",quare", in S1~, the domain of analyticity of X in S1+, then there exi",~ a unique reproducing kernel Pontryagin "'pace P with index v of k x 1 vector valued function$ which are analytic in S1+. Moreover, Aw(-\) u the reproducing kernel ofP and {Awv: w E S1+

and

v E CC k }

Alpay and Dym

40

is dense in P.

PROOF. See e.g., Theorem 6.4 of [AD3].

•

Schwartz [Sch] and independently (though later) Sorojonen [So] were the first to establish a 1:1 correspondence between reproducing kernel Pontryagin spaces of index v and kernels with v negative squares. We shall, as we have already noted, refer to the reproducing kernel Pontryagin space P with reproducing kernel given by (3.1), whose existence and uniqueness is established in Theorem 3.1, as SeX). THEOREM 3.2. If X is a k x m matrix valued function in S1+ which is (S1+, J, p)v admissible and if f belongs to the corresponding reproducing kernel Pontryagin space SeX) and W E S1~, then, for j = 0,1, ... and every choice of v E a:: k ,

belongs to SeX) and (')

If, A,J

vectors

PROOF. in

VI, ... , Vn

(3.2)

vjB(X)

By definition there exists a set of points wI. . .. , Wn in S1~ and such that the n x n Hermitian matrix P with ij entry

a:: k

has v negative eigenvalues A}, ... , Av. Let Ul, •• corresponding to these eigenvalues and let

• , Uv

be an orthonormal set of eigenvectors

j=l,oo.,II.

Then, for any choice of constants

not all of which are zero,

C}, ••• , Cv

v

=

LAilcil2
.)JX(w)* Pw(A)

(3.7)

for every choice of wand A in f!+.

H;

PROOF. Since r is a bounded selfadjoint operator on the Hilbert space it admits a spectral decomposition: r = J~oo tdEt with finite upper and lower limits. Let

r-

=

1:00 tdEt

and

r+ =

100

tdEt .

44

Alpay and Dym

Then

r _ and r + are bounded selfadjoint operators on

H:,

It now follows readily that

'R.r

=

'Rr + [+I'R.r _

is a Krein space since the indicated sum decomposition is both direct and orthogonal with respect to the indefinite inner product [ , Ir given in (3.5), and 'R.r ± is a Hilbert space with respect to ±[ , Ir. It remains to show that

(1)

Awv E C(X) and

(2)

[f, Awvlr = v* f(w)

for every choice of v E a:;k, wE fl+ and

f

E C(X). The identification

= MxJX(w)*.3:.... , Pw

which is immediate from Lemma 3.1, serves to establish (1). Suppose next that f = rg for some g E

H:.

Then

=v*f(w). This establishes (2) for f E limiting argument. •

'R.r.

The same conclusions may be obtained for

f

E

'R.r

by a

THEOREM 3.4. If X is a k x m matrix valued function which is (fl+, J,p)v admissible and if also the multiplication operator M X is bounded on H;', then

8(X) = C(X) .

PROOF. Under the given assumptions, both 8(X) and C(X) are reproducing kernel Pontryagin spaces with the same reproducing kernel. Therefore, by the results of Schwartz [Schl and Sorojonen [Sol cited earlier, or Theorem 3.1, 8(X) = C(X). •

n+

LEMMA 3.2. Let X be a k x m matrix valued function which is analytic in such that Mx is a bounded linear operator from H;J' to H:. Then

(3.8)

Alpay and Dym

45

for j = 0,1, ... and every choice of v E

a:: k

and wE

n+.

PROOF. It is convenient to let .

D} Then, for every choice of u E

oi = --. Ow*}

a:: m

v

and

f=-· Pw

and a E n+,

= Di {v* X(W)u}*

Po«w) = u*DiX(w)*_v_

Pw(o)

= u* Di(M'Xf)(o) ,

where the passage from line 2 to line 3 is just a special case of (3.2) applied to the Hilbert Therefore, space

H:.

as claimed. and

r

•

COROLLARY 1. If X,W and v are as in the preceding lemma and if Aw(.X) are as in (9.1) and (9.9), respectively, then oiv

r--.ow*} Pw

=

{)i

v

--.row*} Pw

oi = --.Awv. ow*}

(3.9)

COROLLARY 2. If X, wand v are as in the preceding lemma and if fi

= J M'X'Pw,iv

,

j = 0,1, ...

,

then li()..) = J

i X(i-s)(w)* L (._ )1 'Pw,s()..)v s=o J s.

(3.10)

and

* Jx(n-l)(w)*v] [/0'" In-I] = [ JX(w) v ... ct»w n (n -I)!

'

.

(3.11)

46

Alpay and Dym

PROOF. Let ni =

ai jaw*i. fi

Then, by Lemmas 3.2 and 3.1,

1 niJ M xV'w,ov * = ]!

=

~ni JX(w)*.!.

J.

Pw

which, by Leibnitz's rule, is readily seen to be equal to the right hand side of (3.10). Formula (3.11) is immediate from (3.10) and the definition (2.4) of ~w,n. • Suitably specialized versions of formulas (3.8)-(3.10) playa useful role in the study of certain classes of matrix polynomials; see Section 11 of [D1] and Section 6 of

[D4]. LEMMA 3.3. If X is as in Lemma 9.2 and if f = J MxV'O/,iu and 9 = J MxV'p,iV for some choice of n, f3 in n+ and u, v in (Vk, then (3.12)

[Mxf,Mxg]8(X) = {Jf,g)H p .

PROOF. This is a straightforward calculation based on the definitions:

= (rV'O/,i u , V'p,i V) Hp = {MxJMxV'O/,iu,V'p,iV)Hp = (Jf,g)H p .

•

Formula (3.12) exhibits Mx as an isometry from the span M of fi = JMxV'O/,ju, j = 0, ... , n in H; into 8(X). This is an important ingredient in the verification of the orthogonal direct sum decomposition

(3.13)

8(X) = 8(X8)[+]XK:(8)

which holds when M = K:(8). In fact M = K:(9) when M is a nondegenerate subspace of Hp,J, as follows from Theorems 2.2 and 2.l.

THEOREM 3.5. Let X = [F G] be a k x m matrix valued meromorphic function on n+ with components F('\) E (Vkxp and G(.\) E (Vkxq such that Mx is a bounded operator from H';: into H;. Then X is analytic on n+ and the following are equivalent:

(1) X is (n+,J,p) admissible.

(2) (9)

r

= MFM} - MaMa is positive semidefinite on

There is a p Ma=MFMs.

X

H;.

q matnz valued analytic function S on

n+ such that IIMslI

~

1 and

47

Alpay and Dym

PROOF. X is analytic on

n+ because v k Mx- E Hp a

for every choice of v E

<em.

The equivalence of (1) and (2) is an easy consequence of the evaluation

* (wi)Vj = [Vj ViAwj r-, PWj

r Vi- ]

PWi 8(X)

Vj vi ) = (r - , - H PWj PWi P

and the fact that finite sums of the form

~Awj Vj

are dense in 8(X).

Suppose next that (2) holds. Then, by a (slight adaptation - to cover the case p", q of a) theorem of Rosenblum [Ro], there exists an operator Q from H3 to H: such that: where, in the last item Ms denotes the (isometric) operator of multiplication by s = b/a on regardless of the size of the positive integer r. Thus, by Theorem 3.3 of [AD4], Q = Ms for some p x q matrix valued function S which is analytic on n+, as needed to complete the proof that (2) => (3). The converse is selfevident. •

H;,

A more leisurely exposition of the proof of this theorem for Pw(A) = 1 - Aw* may be found e.g., in [ADD].

4. RECURSIVE EXTRACTIONS AND THE SCHUR ALGORITHM In this section we study decompositions of the form (3.13) of the reproducing kernel Pontryagin space 8(X) based on a k x m (n+, J,p)v admissible matrix valued function X; the existence and uniqueness of these spaces is established in Theorem 3.l. Such decompositions originate in the work of de Branges [dB2, Theorem 34], [dB3], and de Branges and Rovnyak [dBR] for the case v = 0 (which means that 8(X) is a Hilbert space) and Pw(A) = -21ri(A - w*) for a number of different classes of X and S. Decompositions of the form (3.13) for finite v (i.e., when 8(X) is a reproducing kernel Pontryagin space) and the two cases Pw(A) = I - Aw* and Pw(A) = -21ri(A - w*) were considered in [AD3]; such decompositions in the Krein space setting were studied in [AI] and, for Hilbert spaces of pairs, in [A4]. If B(X) is a nonzero Hilbert space, then it is always possible to find a one dimensional Hilbert space K: (81) such that X K:( 8 t} is isometrically included inside 8 (X). This leads to the decomposition

8(X) = 8(X9t} EB XK:(9t} .

48

Alpayand Dym

Then, if 8(Xet) is nonzero, there is a one-dimensional Hilbert space K(e2) such that X8tK(82) sits isometrically inside 8(Xed, and so forth. This leads to the supplementary sequence of decompositions

8(Xf>t)

= 8(Xele2) EB XetK(e2)

8(Xete2) = 8(Xete2ea)EBXf>te2K(ea)

which can be continued as long as the current space 8(X8t ... en) is nonzero. In this decomposition, the i are "elementary sections" with poles (and directions) which are allowed to vary with j.

e

The classical Schur algorithm corresponds such a sequence of decompositions for the special case in which Pw( A) = 1- AW*, X = [1 S) with S a scalar analytic contractive function in ]I) and all the ei have their poles at infinity. For additional discussion of the Schur algorithm from this point of view and of decompositions of the sort considered above when 8(X) is a reproducing kernel Pontryagin space, see [AD3). In particular, in this setting it is not always possible to choose the K(ei) to be one-dimensional, but, as shown in Theorem 7.2 of [AD3) for the two special choices of p considered there, it is possible to choose decompositions in which the K(ei) are Pontryagin spaces of dimension less than or equal to two. The same conclusions hold for p E Vn also as will be shown later in the section.

THEOREM 4.1. Let X be a k x m matrix valued function which is (f!+, J,p)v admissible and let 8(X) be the (unique) associated reproducing kernel Pontryagin space with reproducing kernel given by (9.1). Let 0' E f!~, the domain of analyticity of X in f!+, let M denote the span of the functions

*

1

ai-t

*v

I

fi = JMX'POI..i-I V = (. -1)1 8w*i- 1 JX(w) , J. Pw W=OI. j

= 1, ... ,n,

(4.1)

endowed with the indefinite inner product

and suppose that the n x n matrix P

= (Pii)

is invertible. Then:

(1)

M is a K(e) space.

(It)

The operator Mx of multiplication by X is an isometry from K(f» into 8(X).

(9)

xe is (f!+, J,p)1'

admissible, where J.I. = v - the number of negative eigenvalues of

P.

(4) 8(X) admits the orthogonal direct sum decomposition 8(X) = 8(Xe)[+JXK:(e) . PROOF. Let

(4.2)

Alpay and Dym

49

denote the m x n matrix with columns JXU-l)(a)*v Vj

=

(j _ I)!

j

'

= 1, .. . ,n

.

Then, by Leibnitz's rule, it is readily checked that

F=[h,···,jn]

= VclIa,n' By (2.4), this can be reexpressed as F(>.) = V{a(>')Aa - b(>.)Ba}-1 ,

where Aa and Ba are as in (2.6). Therefore, by Theorems 2.2 and 2.3, M is a K(8) space. Next, since X Ij

= (.J

1

(j -1)

)' Aa

-1 .

v

in terms of the notation introduced in Section 3, it follows from Theorem 3.2 that X Ii E

8(X) and 1 1 a i - 1 ai-I v*X(>.)JX(w)*vl [Xlj, XIi]8(X) = (i -I)! (j - I)! a>.i-l aw*j-l Pw(A) A=w=a .

(4.3)

But the last evaluation is equal to

as follows by writing j-l k~)

Ij

=L

t=o

-, t. Vj-l-t

with kw(>') = 1/Pw(>') and applying Theorem 3.2 to H~. This completes the proof of

(2). The proofs of (3) and (4) are much the same as the proofs of the corresponding assertions in Theorem 6.13 of [AD3] and are therefore omitted. •

If n = 1, then the matrix P in the last theorem is invertible if and only if X(a)*v is not J neutral. In this case, the 8 which intervenes in the extraction can be expressed in the form 8(>')

= Im + {b a (>') -

l}u(u* Ju)-lu* J ,

where u = JX(a)*v and

bereA) =

s(>.) - sea) 1 - s(A)s(a)*

Alpay and Dyhl

50

with s(..\) = b(..\)/a(..\)j see (2.24) of [AD4J, and hence

v* X(a)S(a) = 0 . Thus it is possible to extract the elementary Blaschke-Potapov factor

from the left of XS to obtain

which is analytic at a and is (Q+,J,p),. admissible, where p. = v ifu* Ju if u· Ju < O.

> 0 and p. =

v-I

THEOREM 4.2. Let X be a k x m matrix valued function which i3 (Q+,J,p)v admi33ible and 3upp03e that 8(X) =I- {O} and Q+ i3 connected. Then X(..\)JX(,,\)* ¢. 0 in nt., the domain of analyticity of X in Q+. any

PROOF. Suppose to the contrary that X(..\)JX(,,\)* == 0 in there exists a h > 0 such that the power series

wEnt.,

Qt..

Then, for

00

X(A) =

L Xs(..\ -

w)S

s=O with k x m matrix coefficients converges, and 00

L

Xs(A _w)sJ(..\* _w*)tx; = 0

s,t=O for 1..\ -

wi < h.

Therefore 00

L

c s+t e i {s-t)6 XsJX; = 0,

s,t=O for 0:5 c < hand 0:5 (J < 271". But this in turn implies that XsJX; = 0 for s,t = 0,1, ... , and hence that X(a)JX({3)* = 0 for la - wi < h and 1{3 - wi < h. Since n+ is connected, this propagates to all of n+ and forces 8(X) = {O}, which contradicts the given hypotheses. Thus X(..\)JX(,,\)* ¢. 0 in as claimed. •

nt.,

COROLLARY 1. If, in the 3etting of Theorem 4.1, n+ i3 connected, then there exi3u a point a E n+ and a vector v E (CAl 3uch that v* X(a)JX(a)*v =I- O. COROLLARY 2. If, in the 3etting of Theorem 4.1, n+ i3 connected, then the 3et of poinu in nt. at which the extraction procedure of Theorem 4.1 can be iterated arbitrarily often (up to the dimemion of 8(X») with one dimen3ional K:(8) 3pace3 i3 deme in n+.

51

Alpay and Dym

If f!+ is not connected, then it is easy to exhibit nonzero 8(X) spaces for which X(,x)JX(,x)* = 0 for every point ,x E f!+. For example, if f!+ has two connected components: f!1 and f!2, let

X(,x) = [a(,x)Kj

and

b(A)Kj] ,

. _ [JpP

J-

o

0]

-Jpp

,x E f!j ,

for

.

Then, for ,x E f!j and we f!i, X(,x)JX(w)* = Pw(,x)KjJppK;

-

Thus X(,x)J X(,x)* space of index p.

{

0 Pw(,x)Ip

if if

i=j ifj·

= 0 for every point ,x E f!+, while 8(X) is a 2p dimensional Pontryagin

THEOREM 4.3. Let X be a k x m matrix valued function which is (f!+, J, p)v admissible such that 8(X) f {OJ and yet X(a)*v is J neutraljor every choice oja E f!~ and v E (I; k, then there exist a pair oj points 0', {3 in f!~ and vectors v, W in (l;k s'IJ.ch that

(1) w*X({3)JX(a)*v

f O.

(£) The two dimensional space M

= span {JX(a)*v POt

, JX({3)*w} P/3

endowed with the J inner product in H;;' will be a x:(e) space. (S) The space 8(X) admits the direct orthogonal sum decomposition 8(X) = 8(Xe)[+]Xx;ce) .

(4) xe is (f!+,J,p)v-1 admissible. PROOF. The proof is easily adapted from the proof of Theorem 7.2 in [AD3] .

•

There is also a nice formula for the e which intervenes in the statement of the last theorem: upon setting 'lJ.l = JX(a)*v, 'lJ.2 = JX({3)*w and

52

Alpay and Dym

we can write

SeA) - s(f3) } ) ( {S(A) - sea) } SeA) = ( 1m + { 1 _ s(A)s(a)* - 1 W12 1m + 1 _ S(A)S(f3)* - 1

W21

)

•

(4.4)

This the counterpart of formula (7.5) of [AD3] in the present more general setting; for more infonnation on elementary factors in this setting, see [AD4]. The reader is invited to check for himself that

J - 8(A)J8(w)* = F(A) Pw(A)

[0,* ,]

-I

0

F( )* w,

where with

A- [a(a) 0

0]* ,

a(f3)

b(a) B = [ 0

0]* ,

b(f3)

and 'Y = UiJu2/p(3(a). The formula

Pw(A)POt(f3)_l {S(A)-S(f3)}{ s(w)-s(a)}* POt(A)Pw(f3) - - 1- s(A)s(a)* 1 - s{w)s(f3)* '

(4.5)

which is the counterpart of (7.3) of [AD3], plays an important role in this verification. The matrix valued function

CIA\ ... c2v-IA\2V-I]

1

with c; =

(~) 2

is (n+,J,p)v admissible with respect to Pw(A) = 1 - AW* and

J = J v+1,v. The corresponding space SeX) is a 2v dimensional Pontryagin space (of polynomials) of index v with reproducing kernel Aw(.X) = (1 - AW*)2v-I. It does not isometrically include a one dimensional K:(8) space such that X8 is (n+, P, J)v-I admissible, since X(A)JX(A)* > 0 for every point A E n+ = ID. This serves to illustrate the need for the two dimensional K:(8) sections of Theorem 4.3. We remark that Theorem 4.1 can be extended to include spaces M in which the chain I;, j = 1, ... , n, is based on a point a E no such that Aw (A) is jointly analytic in A and w* for (A, w) in a neighborhood of the point (a, a) E no x no. In this neighborhood Aw(A) admits a power series expansion (Xl

Aw(A) =

L i,;=O

Ai;(A - a)i(w* - a*);

Alpay and Dym

53

with the k x k matrix coefficients Aij. Now if

v* Aoov P = [

v*AO,n-l V

: V*An-l,OV

...

1

v* An-:l,n-l v

and if Aa and Ba are defined as in (2.6), and

v =

[JX(a)*v", Jx(n-l)(a)*v] (n-1)! '

then A~P Aa - B~P Ba = V* JV .

(4.6)

Formula (4.6) may be verified by differentiating

{a(A)a(w)* - b(A)b(w)*}Aw(A) = X(A)JX(W)* i times with respect to A, j times with respect to w* for i, j = 0, ... , n - 1 and then evaluating both sides with A = w = a. Therefore, by Theorem 2.3, the span M of the columns of F(A) = V{a(A)Aa - b(A)Ba}-1

endowed with the indefinite inner product

is a K:(E» space, whenever the hypothesis of Theorem 2.3 are satisfied. It is important to hear in mind that the indefinite inner product M is now defined in terms of derivatives of Aw(A) and not in terms of evaluations inside Hp which are no longer meaningful since the columns of F (which are just the Ij of (4.1)) do not belong to H;J" when a E no. Nevertheless formula (4.3) is still valid (as follows from the remarks which follow the proof of Theorem 3.2) and serves to justify the assertion that Mx maps K:(E» isometrically into 8(X) in this case also. Recall that a subspace M of an indefinite inner product space is said to be nondegenerate if zero is the only element therein which is orthogonal to all of M. It is readily checked that if M is finite dimensional with basis II, ... , In, then M is nondegenerate if and only if the corresponding Gram matrix is invertible.

THEOREM 4.4. Let P be a reproducing kernel Pontryagin space of k x 1 vector valued functions defined on a set 6. with reproducing kernel Lw{A). Suppose that M = span{Lau}, . .. , Laun} is a nonderenerate .,.,,,b,,pace of P for some choice of a E 6. and Ul," ., Un in a:: k , let [Ul ..... un] denote the k x n matrix with. columm U}, . . . . Un. Th.en:

.N = M[.l and let U =

Alpay and Dym

54

(1)

The matrix U* L OI ( a)U is invertible.

(~)

The sum decomposition P=M[+j...v is direct as well as orthogonal.

(9)

Both the spaces M and...v are reproducing kernel Pontryagin spaces with reproducing kernels (4.7) and (4.8)

respectively. PROOF. The matrix U* LOI(a)U is the Gram matrix of the indicated basis for M. It is invertible because M is nondegenerate. The fact that M is nondegenerate further guarantees that M = {O} and hence that assertion (2) holds.

n...v

Next, it is easily checked that L~(oX), as specified in formula (4.7), is a reproducing kernel for M and hence that M is a reproducing kernel Pontryagin space. Finally, since Lwu - L~u belongs to...v for every choice of w E ~ and u E a:;k and

(J,Lwu - L~uj"p = [f,Lwuj"p = u* few) for every f E ...v, it follows that ...v is also a reproducing kernel Pontryagin space with reproducing kernel L~(oX). • THEOREM 4.5. If, in the setting of Theorem

4.4, P = 8(X) and

Aw(oX) = X(oX)JX(w)* Pw(.A) for some choice of P E Vn and k x m matrix valued function X which is (n+, J, P)II admissible, and if a E n+ (the domain of analyticity of X in n+), then there exists an m x m matrix valued function 8 E p:;(n+) for some finite v such that

(1) M = XK(8). (!) ...v = 8(X8).

(9) L~(oX)

=X(oX) { J-e~:)(~)(w)*} X(w)*.

(4) A~(oX)

= X(oX)8(oX)J8(w)* X(w)* / Pw(oX).

PROOF. To begin with, let

v

= JX(a)*U

and

F(oX) = V POI(oX)-1 .

Then, upon setting A = a(a)* In

and

B

= 6(01)* In

,

55

Alpay and Dym

F can be expressed in the form

F()")

= V{a()")A -

b()")B}-1 ,

which is amenable to the analysis in Section 4 of [AD4J. The latter is partially reviewed in Section 2. In particular, hypotheses (1) and (2) of Theorem 2.1 (above) are clearly met: (1) holds for any /l E no for which la(/l)1 = Ib(/l)1 i- 0 since 0 rf. no, whereas (2) holds because M (which equals the span of the columns of X F) is nondegenerate by assumption. Thus since V*JV P:=--

Po-(o)

is a Hermitian invertible solution of the matrix equation A *P A - B* P B = V* JV ,

it follows from Theorem 2.1 that the span F of the columns of F endowed with the indefinite inner product [Fu, FVJF

= v* Pu

is a K(8) space. This proves (1) and further implies that F(),,)P-1 F(w)

=

J - 8()")J8(w)*

Pw()..) Thus AM()..) = X()..)V {v* JV w

Po-()..)

Po-( 0)

}-l

V*X(w)*

Po-(w)

= X()")F()..)p-1 F(w)* X(w)*

,

which proves (3). The remaining two assertions follow easily from the first two.

•

We remark that if the matrix U which appears in the statement of Theorem 4.4 is invertible, then (4.9) and

(4.10) Conclusion (4) of the last theorem exhibits the fact that (whether U is invertible or not) A~()..) has the same form as Aw()..). Fast algorithms for matrix inversion are based upon this important property. Lev-Ari and Kailath [LAKJ showed that if a kernel Aw()..) is of the form Aw()..) = X()")JX(w)*

Pw()..) for some Pw()..) with Pw()..)* = p.x(w), then the right hand side of (4.10) will be ofthe same form if and only if Pw()..) admits a representation of the form (1.2). The present analysis

56

Alpay and Dym

gives the geometric picture in terms of the reproducing kernel spaces which underlie the purely algebraic methods used in [LAK]. We complete this section with a generalization of Theorem 3.5. THEOREM 4.6. Let X = [C D] be a k x m matriz l1alued function which il

(0+,1, p)" admillible and for which Mx iI a bounded operator from H;:a into H:. Then there eziltl a p X q matriz l1alued meromorphic function 8 on 0+ luch that (1) [Ip - S] iI (O+,J,p)" admillible, and

(£) D = -C8. PROOF. IT v = 0, then the assertion is immediate from Theorem 3.5. IT v > 0, then, by repeated applications of either Theorem 4.1 or 4.3, whichever is applicable, there exists an m X m matrix valued function 6 E 1'j(O+) which is analytic in 0+ such that X 6 is (0+, J, p) admissible and the multiplication operator Me is bounded on H;:a. The last assertion follows from Theorem 6.1 and the formulas for 6 which are provided in and just after the proofs of Theorems 4.1 and 4.3, respectively. Thus, the multiplication operator Mxe is also bounded on H;:a and so, by Theorem 3.5, there exists a p x q matrix valued analytic function 8 0 on 0+ with IIMso II ~ 1 such that

Therefore,

D(821 8 0

+ 622) =

-C(61180

+ 612) ,

which in tum implies that D=-C8 with The indicated inverse exists in 0+ except for at most a countable set of isolated points because det(62180 + 622) is analytic and not identically equal to zero in 0+. Indeed, since 6 is both analytic and J unitary at any point I' E 00 at which la(JJ)1 = Ib(JJ)1 #:. 0, it follows by standard arguments that 671621 is strictly contractive at I' and so too in a little disc centered at 1'. This does the trick, since every such disc has a nonempty intersection with 0+ (otherwise la(..\)/b(..\) 1 ~ 1 in some such disc with equality at the center; this forces b(..\) = ca(..\) , first throughout the disc by the maximum modulus principle, and then throughout all of since it is connected) and 8 0 is contractive in 0+.

°

Now, let F = [II··· fn] be an m X n matrix valued function whose columns form a basis for X:(6), let Q denote the invertible n x n Hermitian matrix with ij entry

and finally, let Y = IIp - S)

and G

= 611 -

S621 .

57

Alpay and Dym

Then it follows readily from the decomposition Y(A)JY(W)* = Y(A) J - 8(A)J8(w)* Y(w)* p",(A) p",(A) = Y(A)F(A)Q-l F(w)*Y(w)*

+ Y(A) 8(A)J8(w)* X(w)* p",(A)

+ G(A) Ip - S;~~l~o(W)* G(w)*

that the difference between the kernel on the left and the first kernel on the right is a positive kernel. Therefore, for any set of points aI, ... , at in the domain of analyticity of S in 0+ and any set of vectors 1, ... , in CV k, the txt matrices

e

et

l.J = 1, ... , t, are ordered: PI 2: P2. Thus, by the minimax characterization of the eigenvalues of a Hermitian matrix, j = 1, ... ,t ,

in which Aj denotes the j'th eigenvalue of the indicated matrix, indexed in increasing size. In particular, AII+I (P2) 2: 0 and hence the kernel based on S has at most 1/ negative squares. On the other hand, since X is (0+, J, P)II admissible, there exists a set of points fJl, ... , fJr in 0+ and vectors 1/1, ... ,1/r in CV k such that the 1· X r matrix with ij entry equal to

1/i X(fJdJ X(fJj )*1/j = 1/iC(fJi) Ip - S(fJi)S(fJj )* C(fJ.)*1/. P{3j(fJi) P{3j(fJi) 3 3 .has exactly 1/ negative eigenvalues. This shows that the kernel based on S has at least 1/ negative eigenvalues, providing that the exhibited equality is meaningful, i.e., providing that the points fJI. ... , fJr lie in the domain of analyticity of S. But if this is not already the case, it can be achieved by arbitrarily small perturbations of the points fJI,.·., fJr because S has at most count ably many isolated poles in 0+. This can be accomplished without decreasing the number of negative eigenvalues of the matrix on the left of the last equality because the matrix will only change a little since X is analytic in 0+, and therefore its eigenvalues will also only change a little. In particular, negative eigenvalues will stay negative, positive eigenvalues will stay positive, but zero eigenvalues could go either way. This can be verified by Rouche's theorem, or by easy estimates; see e.g., Corollary 12.2 of Bhatia [Bh] for the latter. •

5. 'H.p(S) SPACES

In this section we shall first obtain another characterization of the space 'R.r endowed with the indefinite inner product (3.5) in the special case that r = MxJMx is positive semidefinite. We shall then specialize these results to

X = [Ip

-8]

(5.1)

Alpay and Dym

S8

and

(5.2) where S is a p x q matrix valued function which is analytic in f2+ such that the multiplication operator Ms from to is contractive. The resulting space C([Ip - Sj) will then be designated by the symbol1ip (S).

HZ

H:

THEOREM 5.1. If X is a k x m matrix valued function which is analytic in

f2+ such that the multiplication operator Mx from H;;' to H; is bounded and if

then

(1) C(X) = ran

1

r'2

with norm

where P denotes the orthogonal projection of

(£) ran

r

is dense in ran

1

rI

H; onto the kernel of r.

and

{rg, rh}r = (rg, h}Hp for every choice of 9 and h in H;.

(9) C(X) is the reproducing kernel Hilbert space with reproducing kernel given by (9.1).

(,0 X is (f2+, p,J) admissible. (5) C(X) = 8(X). 1

PROOF. Since ker r = ker n, it is readily checked that II Ilr, as defined in 1 1 (1), is indeed a norm on ran r'2. Moreover, if r'2 fn, n = 1,2, ... , is a Cauchy sequence in 1 ran r I , then (I - P)fn is a Cauchy sequence in the Hilbert space H;, and hence tends to a limit 9 in H; as n 1 00. Therefore, since 1- P is an orthogonal projector, it follows by standard arguments that 9 = lim (I - P)fn = lim (I - p)2 fn = (I - P)g nloo

and hence that

nloo

1

IIr I fn

1

-

rIglir = 11(1 -

P)(fn - g)IIHp

= 11(1 - P)fn - gllHp . 1

Thus ran rI is closed with respect to the indicated norm; it is in fact a Hilbert space with respect to the inner product

Alpay and Dym

59

For the particular choice 9 = v/Pw with v E ([!k and wE fl+, the identity 1

(n f, n

1

g)r

= (I -

1

P)f, n 9)Hp

1

= (nf,g)H p =

V

*( nf)(w), 1

serves to exhibit

fg = XJX(w)* v = Aw v Pw 1

as the reproducing kernel for ran f'2. This completes the proof of (1), since there is only 1

one such space. (2) is immediate from (1) and the fact that ker f'2 = ker fj (3), (4) and (5) are covered by Theorems 3.3, 3.5 and 3.4, respectively. • For ease of future reference we summarize the main implications of the preceding theorem directly in the language of the p x q matrix valued function S introduced at the beginning of this section.

THEOREM 5.2. If S is a p x q matrix valued function which is analytic in fl+ such that the multiplication operator Ms from H$ to H: is contractive and if X and J are given by (5.1) and (5.£), respectively, then:

(1) f

= MXJM'X = 1- MsMs

(£) 'Hp(S) = ran

n

1

with 1

IIf'2 fll'Hp(S) = II(I - P)fllHp , 1

where P designates the orthogonal projection of H: onto ker f'2 . 1

(9) ran f is dense in ran f'2 and (fg, fh)'Hp(s) = (fg, h) Hp for every choice of 9 and h in H:.

(4) 'Hp(S) is a reproducing kernel Hilbert space with reproducing kernel Aw(.\) = Ip - S(.\)S(w)* Pw(.\)

(5.3)

The next theorem is the analogue in the present setting of general p E Do of a theorem which originates with de Branges and Rovnyak [dBR1] for Pw(.\) = 1 - .\w*. in

f

n+

E

H:

THEOREM 5.3. Let S be a p x q matrix valued function which is analytic such that the multiplication operator Ms from H' to H: is contractive and for let K,(J) = sup{ltf + MSgllt-p - IIg/lt-p : 9 E Hpq} •

Alpay and Dym

60

Then 1ip (S)

= {f E Hg:

"-u) < oo}

and

(5.4)

PROOF. Let X and J be given by (5.1) and (5.2), respectively. Then clearly Theorem 5.1 is applicable since

r=MXJMX =I-MsMs ~O. 1

Moreover, since r ~ I, it follows that r2 is a contraction and hence, by Theorem 4.1 of 1 Fillmore and Williams [FW], that I E ran n if and only if II

1

2

2

sup { III + (I - nn)2gllHp -lIgliHp : 9 E HpP}
0 and let

M =span{h, ... ,fv} endowed with the J inner product

Then, with the help of Theorem 2.3, it is readily checked that conditions (1) and (2) of Theorem 2.1 are met. Now let

A = diag{Awl' ... , Awn}

and

B = diag{Bw1 ,· .. , Bwn}

be the block diagonal matrices with entries Awj and BWj of size Then F{A} = V{a{A}A - b(A)B}-l

rj x rj

given by (2.6).

{7.3}

and, by Theorem 2.2,

A*PA - B*PB = V· JV .

(7.4)

Alpay and Dym

70

Therefore, by Theorem 2.1, M is a finite dimensional reproducing kernel Hilbert space with reproducing kernel Kw('x) = J - 6('x)J6(w)*

Pw(A) where

e is uniquely specified up to a right J unitary factor by (2.14).

Moreover, since

it follows easily from (2.14) that 6 is analytic in all of 11 except at the union of the zeros of the functions PWI , ••• , PWn' In particular, 6 is analytic in 11+ and is both analytic and invertible at the special point J.I. E 110 which is used in the definition of 6. Thus 6 is invertible in all of 11 except for an at most countable set of isolated points. Moreover, it is readily checked that Theorem 6.1 is applicable for the current choices of A and B, and hence that the multiplication operator Me is a bounded mapping of H;: into itself. Next, since

is a reproducing kernel for

H;:, it is easily seen that

v *ft () a

= (Jft, J =

- 6J6(a)*v) Hp

POt

(f~) - (Jft, 6J6(a)*v) Hp t, Hp POt

POt

-- v *ft ( a ) - (Jf t , 6J6(a)*v) H ,

POt

p

and hence that v* J6(a)J(Me J Jt}(a) = (MeJ/t, J6(a)* JV)H

POt

p

= (Jf 6J6(a)*v) t, Hp

POt

=0 for every choice of a E 11+ and v E «::m. Therefore, since 6 is invertible except for at most a set of isolated points in 11+, it follows that (the analytic vector valued function)

(7.5) Now, by (6.4), where

1111 --

[e01

and

Alpay and Dym

71

By Theorems 6.1 and 6.3, M\f!l and M\f!2 are both bounded multiplication operators on Thus it is meaningful to write

H;J'.

and therefore, since M(,2 is clearly invertible, it follows from (7.5) that

M(,Jft = O. But this in turn reduces to the pair of constraints

(7.6) and

M:2 {M;19t - ht} = 0 . But now as

M:

2

is invertible (with inverse (Me -I)*) the latter constraint implies that 2

This exhibits 0"1 as a solution to the given interpolation problem, thanks to Lemma 7.1 and Theorem 6.3. This completes the proof of the existence of a solution to the stated interpolation problem when P > o. It remains to show that the interpolation problem is also solvable when P ;::: 0, by passage to limit. To this end, let us first write

P=G-H

with entries corresponding to the decomposition

By assumption G > O. Now, just as in [D2], let us replace the 'lij ill the formulation of the problem by "lij(l the perturbed data,

I

+ £)-2"

with £ >

o.

Then the new Pick matrix, corresponding to

1 Pe =G---H 1+£ £ 1 =--G+--P 1+£ 1+£

>_£-G>O -1+£ for every choice of £ > o. Therefore, by the preceding analysis, there exists a p x q matrix valued analytic function Se with IIMsc II ~ 1 such that

Alpay and Dym

72

for j = 1, ... , /I. The proof is completed by letting e ! 0 and invoking the fact that the unit ball in the set of bounded operators from H: to H$ is compact in the weak operator topology (see e.g., Lemma 2.3 on p.102 of [Be]). This means that there exists a contractive operator Q from H: to H$ such that

for every choice of f E H: and g E H$ as e ! 0 through an appropriately chosen subsequence (which we shall not indicate in the notation). In particular,

= (ht,U)H

p •

Now let T = Ms denote the operator of multiplication by s = bl a on H;, regardless of the size of r. Then since MScT = TMSc for every e > 0, it is readily checked that Q*T = TQ* and hence, by Theorem 3.3 of [AD4], that there exists a p x q matrix valued function S which is analytic in n+ such that Q* = Ms. This completes the proof, since S is a solution to the one-sided interpolation problem. • THEOREM 7.2. If the /I x /I matrix P with entries given by (7.2) is positive definite and if is specified by (2.14) for the space M considered in the proof of the last theorem, then

e

{Ta[So]: So is analytic in

n+

and

IIMsoil

$;

I}

is a complete list of the set of solutions to the given interpolation problem. PROOF. The proof is divided into steps. STEP 1. If S is a solution of the interpolation problem, then

Y = [Ip

- SIS

is (0+, J,p) admissible. PROOF OF STEP 1. The argument is adapted from the proof of Theorem 6.2 in [D2]. Let X = [Ip - S] and let

Aw(A) = X(A)JX(W)* Pw(A) denote the reproducing kernel for 'Jtp(S). Then it is readily seen that

Y(a)JY(,B)* = _X~(a...:...)J....,..X-:"(::...,B~)* p~(a)

p~(a)

X(a){J - 9(a)JS(,B)*}X(,B)* p~(a)

= Ap(a) - X(a)F(a)p-lF(,B)* X(,B)*

Alpay and Dym

73

for a and j3 in f2+. Thus if n

f = LAat~t t=1 for any choice of aI, ... ,an in f2+ and (}, ... ,~n in (Vm and if 'Yij denotes the ij entry of p-l, then

n

v

- L L ~;X(as)li(ashij/j(at)* X(at)*~t . s,t=1 i,j=1 Moreover, since S is an interpolant, it follows from Lemma 7.1 and Theorem 5.4 that X Ii E Ji p ( S) and hence that the right hand side of the last equality is equal to v

n

Ilfll~p(S) - L

L (X/;,Aas~shlp(S)'Yij(Aat6,Xfjhlp(S) s,t=1 i,j=1 v

= Ilfll~p(S)

-

L (X/;,fhtp(S)'Yij(f,X/jhlp(S) . i,j=1

But this in turn is equal to

Ilfll~p(S) - IIIIfll~p(S) in which II denotes the orthogonal projection of f onto the span of the Xli in B(X). This last evaluation uses the fact that Mx is an isometry from K(0) into B(X), as noted in the remark following Theorem 5.4. The rest is selfevident since

STEP 2. If S is an interpolant, then

[Ip

- Sl0

= ClIp

- Sol

where C and So are analytic in f2+, and Me is bounded and

IIMsJ ::; 1.

PROOF OF STEP 2. This is immediate from Theorem 3.5 since My is bounded (because both Ms and Me are; the former by assumption and the latter by Theorem 6.1), and Y is (f2+, J,p) admissible, by Step l. for some p

STEP 3. If S is a solution of the given interpolation problem, then S = Te[Sol matrix valued function So with IIMSo II ::; 1.

xq

PROOF OF STEP 3. By Step 2,

011 - Se21 = C

Alpay and Dym

74 and Therefore

S(821So

+ 822) = 811 So + 812

,

which is the same as the asserted formula. The term 821 So + 822 is invertible in n+ by standard arguments which utilize the fact that 8 is J contractive and So is contractive, both in n+. STEP 4. If S = Ta[So] for some pxq matrix valued function So with 1, then S is a solution of the given interpolation problem. PROOF OF STEP 4. By Theorem 6.2, 7.1 it is shown that 0"1 = Ta[O]

IIMsll :5 1.

IIMso II :5

In the proof of Theorem

is a solution of the interpolation problem. This means that

for t = 1, ... , v, and hence that S is a solution of the given interpolation problem if and only if for t = 1, ... , v. But now as

(Ms - M;1)9t = M:;lM(lq+tT2S0)-lMSoM:19t

=0, by (7.6). The calculation is meaningful because all of the indicated multiplication operators are bounded thanks to Theorem 6.3. This completes the proof of the step and the theorem. •

8. REFERENCES [Ab]

M.B. Abrahamse, The Pick interpolation theorem for finitely connected domains, Michigan Math. J. 26 (1979), 195-203.

[AI]

D. Alpay, Krein spaces of analytic functions and an inverse scattering problem, Michigan Math. J. 34 (1987), 349-359.

[A2]

- , Some remarks on reproducing kernel Krein spaces, Rocky Mountain Journal of Mathematics, in press.

[A3]

- , Some reproducing kernel spaces of continuous junctions, J. Math. Anal. Appl. 160 (1991),424-433.

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[A4]

75

- , On linear combinations of positive functions, associated reproducing kernel spaces and a non-Hermitian Schur algorithm, Archiv der Mathematik (Basel), in press.

[AD1]

D. Alpay and H. Dym, Hilbert spaces of analytic functions, inverse scattering, and operator models I, Integral Equations and Operator Theory 7 (1984), 589641.

[AD2]

- , Hilbert spaces of analytic functions, inverse scattering, and operator models II, Integral Equations and Operator Theory 8 (1985), 145-180.

[AD3]

- , On applications of reproducing kernel space", to the Schur algorithm and rational J unitary factorization, in: I. Schur Methods in Operator Theory and Signal Processing (I. Gohberg, ed.), Operator Theory: Advances and Applications OTI8, Birkhauser Verlag, Basel, 1986, pp. 89-159.

[AD4]

- , On a new class of reproducing kernel spaces and a new generalization of the Iohvidov law"" Linear Algebra Appl., in press.

[AD5]

, On a new class of structured reproducing kernel spaces, J. Functional Anal., in press.

[ADD]

D. Alpay, P. Dewilde and H. Dym, On the existence and convergence of solutions to the partial lossle",s inverse scattering problem with applications to estimation theory, IEEE Trans. Inf. Theory,35 (1981), 1184-1205.

[Ar]

N. Aronszajn, Theory of reproducing kernel"" (1950), 337-404.

[AI]

T.Ya. Azizov and I.S. Iohvidov, Linear Operator", in Spaces with an Indefinite Metric, Wiley, New York, 1989.

[Ba1]

J.A. Ball, 235-254.

[Ba2]

- , Factorization and model theory for contraction operators with unitary part, Memoirs Amer. Math. Soc. 198, 1978.

[Be]

B. Beauzamy, Introduction to Operator Theory and Invariant Subspaces, North Holland, 1988.

[Bh]

R. Bhatia, Perturbation bounds for matrix eigenvalues, Pitman Research Notes in Math., 162, Longman, Harlow, UK, 1987.

[Bo]

J. Bognar, Indefinite Inner Product Spaces, Springer-Verlag, Berlin, 1974.

[dB1]

L. de Branges, Hilbert "'paces of analytic functions, I, Trans. Amer. Math. Soc. 106 (1963), 445-468.

[dB2]

- , Hilbert Spaces of Entire Functiona, Prentice Hall, Englewood Cliffs, N.J. . 1968.

[dB3)

- , The expansion theorem for Hilbert spaces of entire functions, in: Entire Functions and Related Topics of Analysis, Proc. Symp. Pure Math., Vol. 11, Amer. Math. Soc., Providence, R.I., 1968.

Trans. Amer. Math. Soc. 68

Models for non contractiona, J. Math. Anal. Appl. 52 (1975),

76

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[dB4)

- , Complementation in Krein spaces, Trans. Amer. Math. Soc. 305 (1988), 277-291.

[dB5)

- , Krein spaces of analytic functions, J. Functional Anal. 81 (1988), 219-259.

[dBR)

L. de Branges and J. Rovnyak, Canonical models in quantum scattering theory, in: Perturbation Theory and its Applications in Quantum Mechanics (C. Wilcox, ed.), Wiley, New York, 1966, pp. 295-392.

[dBS)

L. de Branges and 1. Shulman, Perturbations of unitary transformations, J. Math. Anal. Appl. 23 (1968), 294-326.

[DLS)

A. Dijksma, H. Langer and H. de Snoo, Characteridic functions of unitary operator colligations in 'Irk spaces, in: Operator Theory and Systems (H. Bart, I. Gohberg and M.A. Kaashoek, eds.), Operator Theory: Advances and Applications OT19, Birkhauser Verlag, Basel, 1986, pp. 125-194.

[D1)

H. Dym, Hermitian block Toeplitz matrices, orthogonal polynomials, reproducing kernel Pontryagin spaces, interpolation and extension, in: Orthogonal MatrixValued Polynomials and Applications (I. Gohberg, ed.), Operator Theory: Advances and Applications OT34, Birkhauser Verlag, Basel, 1988, pp. 79-135.

[D2)

- , J Contractive Matrix Functions, Reproducing Kernel Hilbert Spaces and Interpolation, CBMS Regional Conference Series in Mathematics, No. 71, Amer. Math. Soc., Providence, 1989.

[D3)

- , On reproducing kernel spaces, J unitary matrix functions, interpolation and displacement rank, in: The Gohberg Anniversary Collection (H. Dym, S. Goldberg, M.A. Kaashoek and P. Lancaster, eds.), Operator Theory: Advances and Applications OT41, Birkhauser Verlag, Basel, 1989, pp. 173-239.

[D4)

- , On Hermitian block Hankel matrices, matrix polynomials, the Hamburger moment problem, interpolation and maximum entropy, Integral Equations and Operator Theory 12 (1989), 757-812.

[FW)

P.A. Fillmore and J.P. Williams, On operator ranges, Adv. Math. 7 (1971), 254-281.

[IKL)

I.S. Iohvidov, M.G. Krein and H. Langer, Introduction to the Spectral Theory of Operators in Spaces with an Indefinite Metric, Mathematische Forschung, Vol. 9, Akademie-Verlag, Berlin, 1982.

[LAK)

H. Lev-Ari and T. Kailath, Triangular factorization of structured Hermitian matrices, in: I. Schur Methods in Operator Theory and Signal Processing (I. Gohberg, ed.), Operator Theory: Advances and Applications OT18 (1986), 301-324.

[Nu)

A.A. Nudelman, Lecture at Workshop on Operator Theory and Complex Analysis, Sapporo, Japan, June 1991.

[Ro)

M. Rosenblum, A corona theorem for countably many functions, Integral Equations and Operator Theory 3 (1980), 125-137.

[Sch]

L. Schwartz, Sow espaces hilbemens d'espaces vectoriels topologiques et ooyaw: ~socies (noyauz reproduisants), J. Analyse Math. 13 (1964), 115-256.

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[So]

P. Sorjonen, Pontrjaginrii:ume mit einem reprod'Uzierenden Kern, Ann. Acad. Sci. Fenn. Ser. A Math. 594 (1973), 1-30.

[Y]

A. Yang, A con8troction of Krein 8pace8 of analytic junction8, Ph.D. Thesis, Purdue University, May, 1990.

D. Alpay Department of Mathematics Ben Gurion University of the Negev Beer Sheva 84105, Israel

MSC: Primary 47A57, Secondary 47A56

H.Dym Department of Theoretical Mathematics The Weizmann Institute of Science Rehovot 76100, Israel

78

Operator Theory: Advances and Applications, Vol. 59 © 1992 Birkhiiuser Verlag Basel

THE CENTRAL METHOD FOR POSITIVE SEMI-DEFINITE, CONTRACTIVE AND STRONG PARROTT TYPE COMPLETION PROBLEMS

Mihaly Bakonyi and Hugo J. Woerdeman

In this paper we obtain a new linear fractional parametrization for the set of all positive semi-definite completions of a generalized banded partial operator matrix. As applications we obtain a cascade transform parametrization for the set of all contractive completions of a triangular partial operator matrix satisfying possibly an extra linear constraint (thus extending the results on the Strong Parrott problem). In each of the problems also a maximum entropy principle appears.

1. Introduction. In this paper we establish a new parametrization for the set of all positive semi-definite completions of a given "generalized banded" operator matrix. Before we elaborate on this problem we start with.an important application, namely a generalization of the Strong Parrott problem introduced by C. Foia.!l and A. Tannenbaum. It concerns the following. For 1 :5 i :5 j :5 n let Bij : 'Hj ---+ Ki be given bounded linear operators acting between Hilbert spaces. Further, let also be given the operators

(1.1)

We want to find contractive completions of the following problem:

(1.2)

i.e., we want to find Bij, 1 :5 j < i :5 n, so that B = (B;j)f.j=l is a contraction satisfying the linear constraint BS = T. The introduction of the Strong Parrott problem was a consequence of questions arising in the theory of contractive intertwining dilations (see, e.g., the recent book by C. Foia.!l and A.E. Frazho [10]).

Bakonyi and Woerdeman

79

For the problem (1.2) we derive neccessary and sufficient conditions for the existence of a contractive solution. In case these conditions are met we build a so-called "central completion", a solution with several distingueshed properties. From the central completion we construct a cascade transform parametrization for the set of all solutions. As we mentioned before the above results appear as application of our results on positive semi-definite completions. The (strictly) positive definite completion problem is a well studied subject. The first tesults in this domain were by H. Dym and I. Gohberg in [8]. These results were in many ways generalized by H. Dym and I. Gohberg (see the references in [12]) and I. Gohberg, M.A. Kaashoek and H.J. Woerdeman in [12], [13], [14]. A complete Schur analysis of positive semi-definite operator matrices was given by T. Constantinescu in [7], and these results were later used by Gr. Arsene, Z. CeaUi~escu and T. Constantinescu in [1] in positive semi-definite completion problems. An analysis of positive semidefinite completions in the classes of so-called U· DU- and L* DL-factorable positive semi-definite operator matrices was recently given by M. Bakonyi and H.J. Woerdeman in [3]. The methods in [3] cover the finite dimensional case, but do not extend to the general case described in this paper. In the study of positive semi-definite" generalized banded" completions, we first develop some distingueshed properties which uniquely characterize a so called central completion, a notion that appeared in different settings and with different names in [8], [1], [12] and [3]. Next we present a result on which the rest of the paper is based, namely a linear fractional transform parametrization for the set of all solutions. The coefficients of the transformation are obtained from the Cholesky factorizations of the central completion. This is a generalization of results in [12], where the positive definite case was considered, and of results in [3]. Our paper is organized as follows. In Section 2 we treat positive semi-definite completions and in Section 3 contractive completions. Section 4 is dedicated to the study of a generalized Strong Parrott problem. 2. Positive Semi-Definite Completions. Consider the following 3 x 3 problem:

(2.1 ) where (2.2) By this we mean that we want to find the (1,3) entry Al3 of the operator matrix in (2.1) such that with this choice (and with A3l = Ai3) we obtain a positive semi-definite 3 x 3 operator matrix. Note that the positivity of the 2 x 2 operator matrices in (2.2) implies that

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80

where G I : R(A 22 ) -+ R(A 11 ) and G 2 : R(A 33 ) -+ R(A 22 ) are contractions. For a linear operator A we denote by R(A) its range and by R(A) the closure of its range, and if A ~ 0 then AI/2 is its square root with AI/2 ~ O. Let also for a contraction G : C -+ Ie denote Do = (Ie - G*G)I/2 : C -+ C and 'Do = R(D o ). It was proved in [1] that there exists a one-to-one correspondence between the set of all positive semi-definite completions of (2.1) and the set of all contractions G: 'D02 -+ 'Do~ via

(2.3) With the choice G

= 0 we obtain the particular completion A 13

(2.4)

I/2 = A 11I/2G 1 G2 A33·

We shall call this the central completion of (2.1), referring to the fact that in the operator ball in which A I3 lies (namely the one described by (2.3)) we choose the centre. If F is a positive semi-definite operator matrix it is known that there exist an upper triangular operator matrix V and a lower triangular matrix W such that

F

(2.5)

= V"V = wow.

The factorizations (2.5) are called lower-upper and upper-lower Cholesky factorizations, respectively. Moreover if V and Ware upper (lower) triangulars with F = V"V = W·W, then there exists block diagonal unitaries U : R(V) -+ R(V) and 0 : R(W) -+ R(W) with UV = V and Ow = W. This implies that if F is a positive semi-definite n x n operator matrix, then the operators

(2.6)

6. u (F) := diag(l';iVii)i'=I

and

(2.7) do not depend upon the particular choice of V and W in (2.5). Returning to our problem (2.1), if F is an arbitrary completion corresponding to the parameter G in (2.3) then F admits the factorization (2.5) with

AI/2

(2.8)

V

=

(

11

0

o

l/2 G 1 A 22 (G I G2 + Do~ G D02 )A;~2 ) I/2 (Do, G - GjGD )A;~2 D0, A 22 2 02 o DoD02A;~2

and

o

(2.9)

I/2 D O A 22 2 G * AI/2 2

22

Bakonyi and Woerdeman

81

Further, using relations like GiCDaj) ~ Va" one easily obtains that R(V;i) ~ R(V;i) and R(Wii) ~ R(Wii), for all i and j. The triangularity of V and W now yields

(2.10)

-

'R.(V)

1/2 = 'R.(An ) EB Val

-

EB Va, 'R.(W)

= V a * EB Va,

-

1/2

EB'R.(A33 ).

One immediately sees from these equalities that when G = 0 the closures of the ranges of the Cholesky factors of the completion are as large as possible. Relation (2.5) implies the existence of a unitary U : R(W) ~ R(V) with UW = V. A straightforward computation gives us the explicit expression of U, namely

(2.11)

Note that the (3,1) entry in U is zero if and only if G = O. As it will turn out, this will be a characterization for the central completion, thus providing a generalization of the banded inverse characterization in the invertible case, discovered in [8]. We will state the result precisely in the n x n case. Before we can do this we have to recall the following. We remind the reader of the Schur type structure of positive semi-definite matrices obatined in [7]: There exists an one-to-one correspondence between the set of positive semi-definite matrices (Aij )7,j=1 with fixed block diagonal entries and the set of all upper triangular families of contractions 9 = {fiih:5i:5i:5n, where fii = I1?(Aii)' i = 1, ... , n, and f ij : Vr'+I.J ~ Vr,.J-1 for 1 :5 i < j :5 n. The family of contractions 9 is referred to as the choice triangle corresponding to (Aii )7,i=I' In [7] it is also proven that if A ;::: 0 and 9 = {fij h j or (i,j) f/. Sand 0 = W;QVc. Since UQ is strictly upper triangular, we can define

v,;

G

= Q(I -

UQ)-l,

which will give that 0 = W;(I + GU)-lGVc. Since F = Fe - 0 - W, and taking into account (2.32) we obtain that F = T(G). Since F = T(G) is positive semi-definite, the relation (2.20) implies that G is a contraction. This finishes our proof. n 3. Contractive Completions. Consider the following 2 x 2 problem:

(3.1 ) where

Note that the contractivity of the latter operator matrices implies that

where G 1 and G 2 are contractions. It was proved in [2] and [9] that there exists a one-to-one correspondence between the set of all contractive completions of (3.1) and the set of all contractions G : V G, -+ VGj given by

(3.2) With the choice G = 0 we obtain the particular completion BI2 = -G 1 B~I G2 . We shall call this the central completion of (3.1). Let {Bij , 1 ~ j ~ i :5 n} be a n x n contractive triangle, i.e., let Bij : K j -+ Hi, 1 ~ j ~ i :5 n, be operators acting between Hilbert spaces with the property that

In order to make a contractive completion one can proceed as follows: choose a position (io,jo) with io = jo-l, and choose Bio.io such that (Bij)~io:j=l is the central completion

88

Bakonyi and Woerdeman

of {Bij,i ~ io,j ~ jo} as in the 2 x 2 case. Proceed in the same way with the thus obtained partial matrix (some compressing of columns and rows is needed) until all positions are filled. We shall refer to Fc as the central completion of {Bii> (i,j) E T}. THEOREM 3.1. Let {Bij, 1 ~ j ~ i ~ n} be a contractive triangle. Let Fc denote the central completion of {Bij , 1 ~ j ~ i ~ n} and let ~c and \II c be upper and lower triangular operator matrices such that

(3.3) Further, let that

WI :

VFc

--+

ft( ~c) and W2 : V F;

--+

R(\II c) be unitary operator matrices so

(3.4) and put

(3.5) Then each contractive completion of {Bij, 1

~

j

~

i

~

n} is of the form

(3.6)

rt

where G = (GiJ"t-I : R(~c) --+ R(\IIc) is a contraction with G iJ" = 0 whenever (i,j) I,}_ T. Moreover, the correspondence between the set of all positive semi-definite completions and all such contractions G is one-to-one. Furthermore, S( G) is isometric (co-isometric, unitary) if and only if Gis. The decompositions of R(~c) and R(\II c) are simply given by

Proof We apply Theorem 2.2 using the correspondence (3.7)

(;.

~) ~ 0 if and only if IIBII ~ 1.

Consider the (n+n) x (n+n) positive semi-definite band which one obtains by embedding the contractive triangle {Bij, 1 ~ j ~ i ~ n} in a large matrix via (3.7). It is easy to check that when applying Theorem 2.2 on this (n + n) x (n + n) positive semi-definite band one obtains

(use Fc·DF; = DF;Fc). It follows now from Theorem 2.1 that (Tc)ij Further, it is easy to compute that

=

0 for i > j.

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Bakonyi and Woerdeman

where we have

and (3.10) We obtain the first part of the theorem from (3.8) and Theorem 2.2. From relation (3.9) one immediately sees that G is an isometry if and only if S(G) is. Similarly, one obtains from (3.10) that G is a co-isometry if and only if S(G) is. This proves the last statement in the theorem. 0 The existence of an isometric (co-isometric, unitary) completion is reduced to the existence of a strictly upper triangular isometry (co-isometry, unitary) acting between the closures of the ranges of iPe and We. Taking into account the specific structures of iP e and We one recovers the characterizations of existence of such completions given in [5] and [1] (see also [3]). REMARK 3.2. We can apply Theorem 2.1 to characterize the central completion. We first mention that for an arbitrary completion F of {Bij , 1 :::; j :::; i :::; n} one can define iP, Wand T analogously as in (3.3), (3.4), and (3.5). The equivalence of (i), (ii) and (iii) in Theorem 2.1 implies that the central completion is characterized by the maximality of diag(~iiiPi;)l'::l or diag(WiiWi;)i=l' This is a so-called "maximum entropy principle". From the equivalence of (i) and (iv) in Theorem 2.1 one also easily obtains that the uppertriangularity of T characterizes the central completion. 4. Linearly Constrained Contractive Completions. We return to the problem (1.2). The next lemma will reduce this linearly constrained contractive completion problem to a positive semi-definite completion problem. The lemma is a slight variation of an observation by D. Timotin [15]. LEMMA 4.1. Let B: 1i -+ IC, S : g -+ 1i and T : g -+ IC be linear operators acting between Hilbert spaces. Then IIBII :::; 1 and BS = T if and only if (4.1)

I S B*) ( S* S*S T*

~

O.

BTl

Proof. The operator matrix (4.1) is positive semi-definite if and only if

(4.2)

(S*) ( S*T S T*) I B (S B*)

=

(0

T _ BS

T* - S* B* ) 1- BB*

~

0,

and this latter inequality is satisfied if and only if IIBII :::; 1 and BS = T. 0 THEOREM 4.2. Let Bij : 1ij -+ lC i, 1 :::; i :::; j :::; n, Si : 1-{ -+ 1ii, i = 1, ... , nand Tj : 1i -+ ICj be given linear operators acting between Hilbe7't spaces and Sand T be as

Bakonyi and Woerdeman

90

in (1.1). Then there exist contractive completions B of {Bij , 1 $ i $ j $ n} satisfying the linear constraint BS = T if and only if S*S - S(i)*S(i) T(i)* - S(i)*B(i)* ) 1- B(i)B(i)* ~0 T(i) _ B(i)S(i)

(

(4.3)

for i = 1, .. , n,where

Bli

(4.4)

B(i)

=

:

. . . BIn) : ,SCi) . ..

Bii

Bin

(

=

( Si ) S:n ,T(')

=

( TI )

:

T.

fori=I, ... ,n. Proof. By Lemma 4.1 there exists a contractive completion B of {Bij , 1 $ i $ j $ n} satisfying the linear constrained BS = T if and only if there exists a positive semi-definite completion of the partial matrix

(4.5)

I 0

0 I

0

?

0 S*2 BI2 B21

S*n BIn B 2n

?

?

Bnn

S*I Bll

0 0

I

Bil

?

Bi2

Bi2

Sn Bin S*S T*I I TI 0 T2

Bin T*2 0

SI S2

Tn

0

? ? B~n T*n

I

0 0

0

I

As it is known, the existence of a positive semi-definite completion of (4.5) is equivalent to the positive semi-definiteness of the principal submatl'ices of (4.5) formed with known entries. This latter condition is equivalent with (4.3). 0 Let us examine the 2 x 2 case a little further, i.e.,

(4.6) The necessary and sufficient conditions (4.3) for this case reduce to

(4.7) and

(4.8)

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91

Assume that (4.7) and (4.8) are satisfied. Similar to Section 3, let G 1 : 'HI -+ 'VBi2 and G 2 : 'VB12 -+ K2 be contractions such that (4.9) Any solution of the constrained problem (4.6) is in particular a solution of the unconstrained problem (the lower triangular analogue (3.1)), and therefore we must have that (use the analogue of (3.2))

(4.10) where r : 'Val -+ 'Va-2 is some _contraction. The equation B 21 S 1 + B 22 S 2 = T2 implies that r is uniquely defined on R(Dal Sd by (4.11)

We define ro : 'Val -+ 'Va; t.o be the contraction defined on n(DalSd as above, and 0 on the orthogonal complement, i.e.,

ro I 'Val e n(DalSd = 0

(4.12)

We let B~~) denote the corresponding choice for B 21 , that is, (4.13) We shall refer to

B12) ( BBn (O) B 21 22

(4.14)

as the central completion of problem (4.6). In the n x n problem (1.2) (assuming conditions (4.3) are met) we construct step by step the central completion of (1.2) as follows. Start by making the central completion of the 2 x 2 problem

~:

B12 ... Bin) ( ) ( Bn ? B22 ... B 2n : Sn

(4.15)

=(

T1 ) T2

and obtain in this way B~~). Continue by induction and obtain at step p, 1 ::; p ::; n - 1, B~~), . .. ,B~?J-1 by taking the central completion of the 2 x 2 problem

B1,,,-1 ( (4.16)

Bu B (0) _

1' 1,,, ?

.

B(O)

B 1"

.1'-1,1'-1 B"-l,,, ? B,,1'

~.

i·

)

Sl S,,-l S1' S"

~

(i, ).

Bakonyi and Woerdeman

92

The final "result Bo of this process is the central completion of the problem (1.2). LEMMA 4.3. Let Bo be a contractive completion of (1.2). Then Bo is the central completion of (1.2) if and only if (4.17)

( I*

S

BO)

S*S T* I Bo T

is the central completion of the positive semi-definite completion problem (4.5). Proof. By the inheritance principle and the way the central completion is defined it suffices to prove the lemma in the 2 x 2 case. Take an arbitrary contractive completion B of (4.6), corresponding to the parameter r in (4.10), say. The lower-upper Cholesky factorization of the corresponding positive semi-definite completion problem is given by

S

B* )

S*S T* T I

(4.18)

,

where (4.19)

V

=

I S B*) ( 0 0 0 o 0 cpo

and cP is lower triangular such that 1- BB* (4.20)

= CPCP·.

It is straightforward to check that

0) .

cP = (DBi.DGj -G2 B12DGj - DG' rGi DG' Dr.

Since for r = ro the operator D¥-. is maximal among all r satisfying (4.11), the lemma follows from the equivalence of (i) and (ii) in Theorem 2.1. 0 THEOREM 4.4. Let Bo be the central completion of the linearly constrained contractive completion problem (1.2) (for which the conditions (4.3) are satisfied). Let p : 'HI (f] 'H2 - n«s* S - T*T)I/2) be such that (4.21)

and Wand cP lower triangulars such that (4.22)

W*W = I - pOp -

B~Bo

and (4.23)

Consider the contraction WI : VBo - 'R.( w) and the unitary W2 : 'R.( cp.) - VB. with the properties (4.24)

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93

and

(4.25) Finally, define

(4.26) Then there exists an one-to-one correspondence between the set of all contractive solutions of the problem (1.2) and the set of all strictly lower triangular contractions G: R.(III) -+ R.(~*) given by (4.27)

V(G)

= Bo -

~(I

+ Grt 1 GIII

Moreover, V (G) is a co-isometry if and only if G is a co-isometry and V (G) is an isometry if and only if S* S = T*T and G is an isometry. The decompositions of R.(~*) and R.(III) are simply given by

R.(~*)

= ffii'=1 R.(~:i)' R.(III) = ffii'=1 R.(lIIii).

Proof. We shall obtain our results by applying Theorem 2.2 for the positive semidefinite completion problem (4.5). Straightforward computation yield that

Vc=

(4.28)

( 0IS0 Bo) 0 o

0

~*

and (S* S - o T*T)I/2 T

(4.29)

o~ )

We remark here that the relation (4.30)

S* S - T*T

= S* D2Bo S > S* D4Bo S -

gives the existence of the contraction p with (4.21). Now we have to determine the unitary U = (Uij)~,j=1 so that UWc = Vc. Note that the existence of WI and W2 is assured by the relations (4.22) and (4.23). An immediate computation shows that

U= ( .umtary . ·h h were WI IS WIt ( WI)

111* 0

p*

-w;BoWj

-w;Bowj

o

Bo) 0 ~*

94

Bakonyi and Woerdeman

Substituting these data in the first equality of (2.20) gives

(4.31 )

T( (

OOG*) 0 0 0 o0 0

)=

(I

S V(G)* ) , S*S T* V(G) T Q(G) ~

where V( G) is given by (4.27) and (4.32)

1= Q(G)

= V(G)V(G)* + ~(I + Gr)-I(I -

GG*)(I + Gr)*-I~*

The first part of the theorem now follows from (4.31) and Lemma 4.1. Further, (4.32) implies that V(G) is a co-isometry if and only if Gis. If the contractive solution V(G) to the constrained problem (1.2) is isometric, then clearly we must have that S* S = T*T and thus p = O. In this case, 111

0 0) .

Wc= ( 0 00 Eo T I

(4.33)

Using the second inequality in (2.20) in this special case, we obtain that 0 0

T( ( 0 0

(4.34)

o

0

G* )

o )= o

S V(G)* ) ( Q(G) T* S* S*S I V(G) T

where (4.35)

1= Q(G)

= V(G)*V(G) + 111*(1 + Gr)-I(1 -

G*G)(I + Gr)*-IIII.

Relation (4.35) implies that when S*S = T*T, the spaces VV(G) and VG have the same dimensions and thus V(G) is isometric if and only if G is. This finishes the proof. 0 In the 2 x 2 case another parametrization was derived in [4]. REMARK 4.5. By Theorem 4.4 we can reduce the existence of a co-isometric completion of the problem (1.2) to the existence of a strictly lower triangular co-isometry acting between R(III) and R(~·). Also, when S*S = T*T, the existence of a isometric completion of the problem (1.2) reduces to the existence of a strictly lower triangular isometry acting between R(III) and R( ~*). REMARK 4.6. There exists a unique solution to (1.2) if and only if 0 is the only strictly lower triangular contraction acting R( 111) --+ R( ~*). This can be translated in the following. If io denotes the minimal index for which lIIioio =I 0, then there exists a unique solution if and only if ~kk = 0 for k = io + 1, ... , n. REMARK 4.7. As in Remark 3.2 the upper triangularity of r characterizes the central completion. For this one can simply use Theorem 2.1 and Lemma 4.3. Also the maximality of diag(~ii~ii)f=I or diag(lIIiillli;)f=I characterizes the central completion (a maximum entropy principle). For a different analysis in the 2 x 2 case we refer to [4] ..

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95

REFERENCES [1) Gr. Arsene, Z. Ceau§escu, and T. Constantinescu. Schur Analysis of Some Completion Problems. Linear Algebra and its Applications. 109: 1-36, 1988. (2) Gr. Arsene and A. Gheondea. Completing Matrix Contractions. J. Operator Theory. 7: 179-189, 1982. (3) M. Bakonyi and H.J. Woerdeman. Positive Semi-Definite and Contractive Completions of Operator Matrices, submitted. (4) M. Bakonyi and H.J. Woerdeman. On the Strong Parrott Completion Problem, to appear in Proceedings of the AMS. (5) J .A. Ball and I. Gohberg. Classification of Shift Invariant Subspaces of Matrices With Hermitian Form and Completion of Matrices. Operator Theory: Adv. Appl. 19: 23-85, 1986. (6) J.P. Burg, Maximum Entropy Spectral Analysis, Doctoral Dissertation, Department of Geophysics, Stanford University, 1975. (7) T. Constantinescu, A Schur Analysis of Positive Block Matrices. in: I. Schur Methods in Operator Theory and Signal Processing (Ed. I. Gohberg). Operator Theory: Advances and Applications 18, Birkhauser Verlag, 1986, 191-206. (8) H. Dym and I. Gohberg. Extensions of Band Matrices with Band Inverses. Linear Algebra Appl. 36: 1-24, 1981. (9) C. Davis, W.M. Kahan, and H.F. Weinberger. Norm Preserving Dilations and Their Applications to Optimal Error Bounds. SIAM J. Numer. Anal. 19: 444-469, 1982. (10) C. Foias and A. E. Frazho. The Commutant Lifting Approach to Interpolation Problems. Operator Theory: Advances and Applications, Vol. 44. Birkhauser, 1990. (11) C. Foias and A. Tannenbaum. A Strong Parrott Theorem. Proceedings of the AMS 106: 777-784, 1989. (12) I. Gohberg, M. A. Kaashoek and H. J. Woerdeman. The Band Method For Positive and Contractive Extension Problems. J. Operator Theory 22: 109-155,1989. (13) I. Gohberg, M. A. Kaashoek and H. J. Woerdeman. The Band Method For Positive and Contractive Extension Problems: an Alternative Version and New Applications. Integral Equations Operator Theory 12: 343-382, 1989. . (14) I. Gohberg, M. A. Kaashoek and H. J. Woerdeman. A Maximum Entropy Priciple in the General Framework of the Band Method. J. Funct. Anal. 95: 231-254, 1991. (15) D. Timotin, A Note on Parrott's Strong Theorem, preprint.

Department of Mathematics The College of William and Mary Williamsburg, Virginia 23187-8795

MSC: Primary 47 A20, Secondary 47 A65

96

Operator Theory: Advances and Applications, Vol. 59 © 1992 Birkhauser Verlag Basel

INTERPOLATION BY RATIONAL MATRIX FUNCTIONS AND STABILITY OF FEEDBACK SYSTEMS: THE 4-BLOCK CASE

Joseph A. Ball and Marek Rakowski

Abstract. We consider the problem of constructing rational matrix functions which satisfy a set of finite order directional interpolation conditions on the left and right, as well as a collection of infinite order directional interpolation conditions on both sides. We set down consistency requirements for solutions to exist as well as a normalization procedure to make the conditions independent, and show how the general standard problem of Hoo control fits into this framework. We also solve an inverse problem: given an admissible set of interpolation conditions, we characterize the collection of plants for which the associated Hoo-control problem is equivalent to the prescribed interpolation problem. Key words: Lumped and generic interpolation, homogeneous interpolation problem, stabilizing compensators, 4-block problem, HOO control.

Introduction The tangential (also called directional) interpolation problem for rational matrix functions (with or without an additional norm constraint) has attracted a lot of interest in the past few years (see [ABDS, BGRI-6, BH, BRan, D, FF, Ki]); much of this work was spurred on by the connections with the original frequency domain approach to Hoo-control theory (see [BGR4, BGR5, DGKF,

Fr, Ki, V]). The simplest case of the interpolation problem is of the following sort. We are given points

Zl, ••• ,ZM

in some subset u of the complex plane

and 1 X n row vectors 111,' .. ,11M and seek a rational m

ee, nonzero 1 X m row vectors Xl, ••• ,XM X

n matrix function W( z) analytic on u

which satisfies

(0.1)

X.W(zt)

= 11.,

i

= 1,,,, ,M.

In the two-sided version of the problem, we are given additional points

n X 1 column vectors

1.11, ••• ,UN

and m X 1 column vectors VI, •••

,VN

WI,' •• ,WN

in u, nonzero

and demand in addition that

W satisfy

(9·2)

W(Wj)Uj

= Vj,

If for some pair of indices (i, j) it happens that zt

j

= 1,,,, ,N.

= Wj, in applications a third type of interpolation

condition arises (0.3)

Z.W'({ij)Uj = Pi; whenever zt =

Wj

=: {ij

Ball and Rakowski

and where

and

r

Pij

97

is a given number. By introducing matrices

= hijh~i~M.l~j~N where "Yij

={

Pij (Wi - Zj)-lZiUj

if Zi = Wj if Zi '" Wj

the conditions (0.1)-(0.3) can be written in the streamlined compact form

(0.1') zoEa'

(0.2') ..DE ..

LRes..=..o(z - Ad-1 B+W(z)C_(z - A"yl

(0.3')

=r

ZoED'

where Res ..= ..oX(z) is the residue of the meromorphic matrix function X(z) at zoo Interpolation conditions of higher multiplicity can be encoded by allowing the matrices A, and A". to have a more general Jordan form. This is the formalism developed in [BGR4). If (T is either the unit disk or the right half plane, one can also consider the problem with an additional norm constraint

sup IIW(z)1I < 1

(0.4)

..E..

to arrive at a matrix version of the classical Nevanlinna-Pick or Hermite-Fejer interpolation problem. In [BGR4) a systematic analysis of the problem (0.1')-(0.3'), with or without the norm constraint (0.4), is presented, including realization formulas (in the sense of systems theory) for the set of all solutions. In [BR5), in addition to the

1Jl.mJu:d interpolation conditions (0.1)-(0.3) or (0.1 ')-(0.3')

there was imposed a generic (or infinite order) interpolation condition

(P+ W)(i)(zo) for some

(0.5)

zo E

(T,

= p~)(zo) for j = 0,1,2""

or equivalently

P+(z)W(z)

= P_(z) for all z,

98

Ball and Rakowski

where P+ and P_ are given polynomial matrices of respective sizes K x m and K x n.

In

[BR5) the theory developed in [BGR4) for the problem (0.1')-(0.3') was extended to handle the

problem (0.1 ')-(0.3') together with (0.5) (with or without the additional norm constraint (0.4», with the exception of the explicit realization formulas for the linear fractional parametrization of the set of all solutions. The features obtained include: an admissibility criterion for data sets

(C+, C_, A,.., A" B+, B_, r, P+(z), P_(z» which guarantees consistency and minimizes redundancy in the set of interpolation conditions (0.1 ')-(0.3') and (0.5), reduction of the construction of the linear fractional parametrizer of the set of all solutions to the solution of a related homogeneous interpolation problem, and a constructive procedure for solving this latter homogeneous interpolation problem. The associated homogeneous interpolation problem is of the following form. In general, we let 'R denote the field of rational functions and 'Rmxn denotes the set of m x n matrices over 'R

/J(~i)

P"(~i)

has full row rank. By conditions (NPDSviii) and (NPDSx), geometric multiplicity of a zero of H in

0',

there is a point

~

AiQ(~;)

=

o.

Since p. is the largest

such that the matrix

A= 4>/J(~) P,.(~)

has full row rank and AQ( ~) = O. Hence we can add to

4>; (j = K. + 1, K. + 2, ... , p.) a function

p(z)c,

where c is a vector and p(z) is a scalar polynomial vanishing at ~t. ~2' .•• ,~. to the order p., so that the modified 4>K+l, 4>,,+2, ••• ,4>/J (which are again called 4>,,+1, ••• ,4>/J) are such that the matrix

Ball and Rakowski

108

Hi=

4>"(.~i)

P,,(Ai)

has full row rank and HiQ( Ai)

= O.

In this manner we obtain functions 4>1,4>2,· .• ,4>,. whose span

is orthogonal to the row span of P" on q( A".) U q( A,) and such that the function

+(z)

=

[~1;ll

Q(z)

4>,.(z) vanishes on q(A".) U q(Ad. Now it follows from the construction and condition (NPDSvili) that if

4>i is a left null function for H at AI. A2,··· ,A. of orders h, 12, ... ,I., then 4>i(Z)Q(Z) vanishes at Ai to the order at least kj where kj

~

max{l,li}. Suppose ki

4>i(Z)Q(Z)

= (.Ii (z 3=1

< 00. By condition (NPDSvii),

Ai)k;)Vi(Z)Q(Z)

= ~i(Z)Q(Z). where Vi is analytic, and does not vanish, at AI, A2, ... ,A•.

Let,pi

= 4>i -

~i (i

= 1,2,··· ,1-').

Then the set {,p1o,p2,··· ,,p,.} contains a canonical set ofleft

null functions for H at each zero of H in span of P" on q(A".) U q(Ad, and ,piQ

q,

the span of {,pI, ,p2, ... ,,p,.} is orthogonal to the row

= 0 (i = 1,2,··· ,1-').

Extend the span of {,p1o ,p2, ... ,,p,.} to an orthogonal complement B of the row span of

P" in (Qol, q(A".) U q(A,», and project each column of W 2 onto an orthogonal complement of the row span of Q in

(P~T,

q( A".) U q( A,» along BOT

+ {column span of Q}

to get W3 •

Step 4 Multiply W3 on the right by a regular rational matrix function without poles or zeros in

q(A".) U q(Ad, so that the resulting function W4 has no zeros nor poles in q \ (q(A".) U q(Ad). Step 5 Find a minimal polynomial basis column span of [W4 Q] in

(P~T,

{UIo U2, .••

,uT } for an orthogonal complement of the

q(A".) U q(Ad). Set

The extended null-pole subspace S,,(w, p .. , Q) given in (2.2) has a special relationship with the null-pole subspace Sew, p .. ) studied in [BRa] as the following result shows.

109

Ball and Rakowski

2.2 .. Suppose (w,PIC(z),Q(z))

= (C".,A".,A"BC,r,PIC(z),Q(z»

is a a-admissible extended null-pole data set and let StreW, PIC) given by (1.5) and StreW, PIC' Q) given by (2.2) be the associated "R( 0' )-modules. Then THEOREM

PROOF:

The containment C is trivial from the definition. Conversely, suppose that j E StreW, PIC' Q)

is analytic on

0'.

By (2,2)

j(z)

(2.3) where x E

rx.

ctnw,h E "RmXl(a), r

E

= c".(z -

A". )-l x + h(z) + Q(z)r(z)

"Rkxl are such that PICj = 0 and

In general, if 9 E "Rmx1 has partial fraction decomposition 9

(elements of "Rmx1 with all poles inside ~c(9)

= 9_. Since C".(z -

0'

E

Resz=z.(z-Ad-1B,h(z) =

z.E"

= 9_ + 9+ where 9_ E "RO'Xl(a c )

and vanishing at infinity) and 9+ E "RmXl(a), we define

A".)-l x E "RO'Xl(aC) and h E "RmXl(a) in (2.2), we have

(2.4) But a consequence of (NPDSix) is that (2.4) can happen only if

From (NPDSi) and Lemma 12.2.2 in [BGR4), we conclude that x

= o.

Hence (2.3) becomes

j(z) = h(z) + Q(z)r(z) where PICj

= 0 and E Res..=z. (z -

A". )-1 BCh(z)

= O.

Since Qr is analytic on

(J'

and by (NPDSvii)

zaEr

Q has no zeros on 0', we must have that r is analytic on 0'. Furthermore, from (NPDSviii) we see that

E Resz=z.(z-Ad-1 BcQ(z)r(z) = o. Then j = h+Qr in fact is an element of StreW, PIC)n"RmXI(a) z.E"

as asserted. 3. Interpolation Problem.

Let

0'

be

&. subset

of the complex plane

ct. We look for a rational matrix function W

which is analytic on/O" and satisfies the following five interpolation conditions. Let

Z1,Z2,··· ,ZM

be given (not necessarily distinct) points in

scribed 1 X m and 1 x n vector functions analytic at zi (j be positive integers. We require that

0'.

Let

Xi

and

Yi

be pre-

= 1,2,··· ,M), and let k 1 , k 2 ,··· , kM

Ball and Rakowski

110

(3.1) for i

= 1,2,··· , kj and j

= 1,2,··· , M.

Let WI, W2, ..• , WN be given (not necessarily distinct) points on

0'.

Let Uj and Vj be

prescribed n x 1 and m x 1 vector functions analytic at Wj (j = 1,2,··· , N) and let It, l2,··· , IN be positive integers. The second condition is that

(3.2) for i

= 1,2,··· ,Ij

and j = 1,2,··· ,N.

For each pair of points Zi and Wj such that Zi

Wj,

we are given numbers /fg(1

1,2,··· , ki and 9 = 1,2,··· , Ij) so that df +1- 1

(3.3)

(J)

-:-(1-=-+-g------:1)-:"! -:"dz--:f:-:-+-g--:-1 (xi

(g)

( z) W ( Z )uj

I

( z)) Z=Zi

= / f g,

where M'1)(z) is a polynomial obtained by discarding all but the first 1/ coefficients in the Taylor expansion of a function h at Zi; that is, if h(z) = L:~1 h{i}(z - Zi)j-l in a neighborhood of Zi, M'1)(z) =

L:']=1 h{i}(z -

Zi)j-l.

The fourth interpolation condition is as follows. We are given 1 x m and 1 x n rational vector functions Pj+ and pj_ (j = 1,2,··· , K) analytic on

0'

and require that

(3.4) for all positive integers i and for at least one (and hence for all) A EO'. Finally, we are given n x 1 and m analytic on

0'

X

1 rational vector functions qj_ and qj+ (j = 1,2,··· , L)

and demand that i 1

dd i-I (W(z)qj_(Z))

(3.5)

Z

I

= ddZ i-I qj+(Z) I Z=A Z=A i 1

for all positive integers i and for at least one (and hence all) points A E 0'. Below, we reformulate the conditions (3.1) - (3.5) and show when they are consistent. The combination (3.1) - (3.4) was considered in detail in [BR5]; we summarize this case (the so-called 2-block case) first. We note that the problem of finding a rational matrix function which is analytic on

0'

and satisfies conditions (3.1) - (3.3) is called the two-sided Lagra.nge-Sylyester interpolation

problem. Its complete solution is presented in Chapter 16 of [BGR4J. The problem of finding a rational matrix function which is analytic on in [BRS].

0'

and satisfies conditions (3.1) - (3.4) has been solved

Ball and Rakowski

111

3.1 2-block Interpolation Problem.

In [BR5] the following general interpolation problem was considered. We are given a subset u of the complex plane a; and an interpolation data set

(3.6) consisting of matrices C+ E a;mxn·,C_ E a;nxn., A". E a;n.xn·,A( E a;n(xn(,B+ E a;n(xm,

B_ E a;n( xn,

r

E a;n( xn. and matrix polynomials P+(z) E nKxm and P_(z) E nKxn which

satisfy the admissibility requirements (IDSi) the pair (C _, A".) is observable and u( A".) C Uj (IDSii) the pair (A(, B+) is controllable and u(Ad C Uj (IDSiii)

r A". - A(r = B+C+ + B_C_

(IDSiv) p_CRnXl) C p+(nmXl), P+(z) has no zeros in

U

and the rows of [P+(z) P_(z)] form a

minimal polynomial basisj (IDSv) the function

is analytic on a;. (IDSvi) if 0"( A".) = {At. A2, ... , Ar }, the pair

j[

P+(Al) B+ ) P+(A2)

ArI

'

P+~Ar)

is controllable. A collection of data w in (3.1.1) satisfying (IDSi) - (IDSvi) is said to be a uadmissible interpolation data set. The problem then is to describe all rational matrix functions with W(z) analytic on u which satisfy the interpolation conditions

(3.1') zoEO'

(3.2') zoEa'

(3.3')

L zoECI

and

(3.4')

Resz=zo(z -

Ad- 1 B+W(z)C_(z - A".)-l

=

r

112

Ball and Rakowski

We mention that the problem (3.1') - (3.4') is simply a more compact way of writing conditions (3.1) - (3.4). Indeed (3.1') is equivalent to (3.1) if we let, for each j

(3.7)

A(j

['

= {

1

Zj

,8;+ =

1

= 1,2, .. · ,M, {l}

[:!:){,) 1

Yj

{2}

,8;_ =-

z{l.j}

Zj

yJI.i}

J

where f{i} denotes the

ith

Yj

coefficient in the Taylor expansion of a function fat

Zj,

and set

(3.8)

Conversely, if matrices A(, B+, B_ of appropriate sizes are given, there exists a nonsingular matrix

S such that SA,S-l is a block diagonal matrix with the diagonal blocks in lower Jordan form. After replacing A( by SA,S-I, B+ by SB+ and B_ by SB_, we can convert the interpolation condition (3.1) to the equivalent condition (3.1'). Similarly, if for each j

= 1,2,··· , N

we let

(3.9)

Wj

A"'j =

[

1 Wj

and then set

A".

= [ A""

A"..

1

. A".N

then (3.2') collapses to (3.2). Conversely, if the matrices C+, C_, A". are given, we can find a nonsingular matrix S such that S-lA".S is in Jordan form. After replacing A ... by S-lA".S, C_ by

C_S, and C+ by C+S, we can reformulate the interpolation condition {3.2'} in the more detailed form {3.2}.

Ball and Rakowski

113

A similar analysis gives an equivalence between (3.4) and (3.4'). Suppose first that and let rii and

Wj

If Zi

f=

Z;

= wi

= hlg] where 'Y!i! = 1,2, .. ·ki and 9 = 1,2,,,, ,Ii) are numbers associated with Zi

as in (3.3). Then the interpolation conditions (3.3) hold if and only if

Wj

let rij be the unique solution of the Sylvester equation in X

where A".j and A" are as in (3.7) and (3.8) (see [BGR4], Appendix A.l) and set r = [fij] with 1 ::::; i ::::; M and 1 ::::; j ::::; N. Then the set of interpolation conditions (3.3) is equivalent to the single

block interpolation condition (3.3'). We also mention that validity of the Sylvester equation

is a necessary and sufficient condition for the consistency of (3.1') - (3.3') (see [BGR4] or [BR5]). Conditions (3.3) can be recovered from (3.3') by multiplying both sides of the equality by nonsingular matrices Sand T such that SA,S-l and T-l A,T are in lower and upper Jordan forms, respectively, and reading the numbers "(Ig from the appropriate blocks ofthe matrix SrT. Finally, if P+ and P_ are rational matrix functions whose jt" rows (j

= 1""

, K) are

equal to Pj+ and -Pj_ respectively, then it is easy to see that conditions (3.4) are equivalent to (3.4'). Demanding that P+ and P_ are polynomials is with no loss of generality. The admissibility conditions (IDSi) - (IDSvi) guarantee consistency and minimize redundancies in the interpolation conditions. For complete details we refer to [BR5]. 3.2 The general 4-block case.

Let Q_ and Q+ be rational matrix functions whose

P"

columns (j

= 1,2"" , L) are

equal to qj_ and qj+, respectively. Conditions (3.5) are equivalent to

(3.5') Similarly as with P+ and P_, we will assume first Q_ and Q+ are matrix polynomials and the columns of

(3.10) form a minimal basis.

Ball and Rakowski

114

We consider now the consistency of condition (3.5'). It follows immediately from (3.5') that

Also, since W is analytic on u, Q_ has no zeros in u. Indeed, if Q_ had a zero in u, then Q+ = W Q_ would have a zero in u, and so the rank of the matrix polynomial (3.10) would drop at some point of u, contradicting the fact that the columns of (3.10) form a minimal polynomial basis. Condition (3.1) says that the first kj Taylor coefficients at Zj of Xj(z)W(z) coincide with the first kj Taylor coefficients at Zj of Yj(z). Therefore, by (3.5'), the first kj Taylor coefficients at

zi of Xj(z)Q+(z) coincide with the first kj Taylor coefficients at Zj of y;(z)Q_(z). Consequently, the function

(z - AC;)-l[Bj+ Bj-l

[3~~~~n

is analytic on CJ: where AC;' Bj+ and Bj_ are as in (3.7). Hence the function

is analytic on CJ:. Suppose that a vector Uj(Wj) is in the column span of Q_(Wj), where functions in (3.2). Then Uj(Wj)

Uj

is one of the

= Q_{Wj)Uo, and it follows from (3.2) and (3.5') that

So in this case conditions (3.2) and (3.5) are either redundant or contradictory. To exclude such situations we require that the matrix

[Ui,(Wj,) uj,(wj,) .. . ui. (wi.) Q-{Wi'>] have full column rank whenever Wj,

= wi. = ... = Wj..

In terms of the data A"., C+, C_, this

assumption is equivalent to the controllability of the pair

(3.11)

where {wi"wi.,··· ,Wi.} is the set of distinct points among {WtoW2,··· ,WN}. We will call this assumption the consistency of right generic and right lumped interpolation conditions. Finally, in view of (3.4') and (3.5'),

Ball and Rakowski

115

P+Q+ = P+WQ_ = -P-Q-. So conditions (3.4') and (3.5') are consistent if

We summarize various consistency and minimality properties our interpolation data must have in the following definition. We will call the data

(3.12) a q-admissible interpolation JiiWl.ill if C+ E a: mxn • , C_ E a: nxn ., A". E a:n• xn., A( E a:ne xne, B+ E

a:nexm,B_ E a:nexn,r E a:nexn·,p+(z) E nKxm, P_(z) E nKxn,Q_ E nmxL,and Q+ E nnxL are such that (IDSi) the pair (C_,A".) is observable and O'(A".) C 0'; (IDSii) the pair (A(, B+) is controllable and O'(A".) C 0'; (IDSiii) r A". - A,r = B+C+ (IDSiv) P_ nnxI

c

+ B_C_;

p+nmxI, P+(z) has no zeros in 0' and the rows of [P+(z) P_(z)] form a

minimal polynomial basis; (IDSv) the function

is analytic on a:. (IDSvi) if 0'( A".)

= {AI, A2, ... , AT}, the pair P+(At) B+ ) P+(A2)

P+(AT) is controllable; (IDSvii) n1xmQ+(z)

c nlxnQ_(z), Q_(z) has

form a minimal polynomial basis;

no zeros in 0', and the columns of

Ball and Rakowski

116

(IDSviii) the function

is analytic on 0:. (IDSix) if u(A".)

= {w}, W2,··· , w.}, then the pair

J)

is observable. (IDSx) [P+ P_]

[~~] = o.

Given a u-admissible interpolation data set, the interpolation problem is to find which m x n rational matrix functions W analytic on u satisfy conditions (3.1') - (3.5'). 4. Parametrization of solutions.

Theorems 4.1 and 4.2 below present our results on parameterization of solutions of the interpolation problem (3.1') - (3.5') (respectively with or without an additional norm constraint). We first need the following observation. Ifw

= (C+, C_, A"., A"

B+, B_, r, P+(z), P_(z), Q_(z),Q+(z»

is a u-admissible interpolation data set, then the collection of matrices

(4.1) together with the matrix polynomials PI
O} or the unit disk V

= {z : Izl


= C_(z -

A". )-lZ +

k_ if k E n.nxl has

and vanishes at infinity

we get that P~c(f-)

= OJ

since

(4.4) But now by condition (IDSix), (4.4) forces

(4.5) By Lemma 12.2.2 in [BGR41, the first condition in (4.5) forces z = O. Also, since by (IDSvii) Q_ has no zeros in

0',

the second condition in (4.5) forces r to be analytic on

h(z) = h+(z) + Q+(z)r(z) is analytic on

0'.

[}~]

But then

Thus S satisfies conditions (ii-a) in Theorem 4.3.

To verify (ii-b) we must show that for any given so that

0'.

f-

E n.mXl(O') we can find hE n.mXl(O')

E S. But by Theorem 2.2, S.,.(w,p)nn.<m+n)xl(O') = S.,.(w,p,Q)nn.<m+n)xl(O').

Hence this result follows in the same way as for the 2-block case (see Theorem 4.3 in [BR5]). Conversely, suppose which S

= S.,.(w,[p+

P-l,

(w, [P+ P-l, [8~]) is a O'-admissible extended null-pole data set for

[8~])

satisfies (ii-a) and (ii-b). Thus (w,[p+ P-l,

[8~])

satisfies

(NPDSi)-(NPDSx) and we must verify that w satisfies (IDSi) - (IDSx). First note that (IDSiii), (IDSv), (IDSviii) and (IDSx) follow from their counterparts (NPDSiii), (NPDSv), (NPDSviii) and (NPDSx) respectively without use of any assumptions on S.,.

(w, [P+ P-l, [8~]). Next, one can

use Theorem 2.3, assumption (ii-b) and the same arguments as those for the 2-block case (see Theorem 4.3 in [BR5]) to deduce (IDSii), (IDSiv) and (IDSvi) from their counterparts (NPDSii), (NPDSiv) and (NPDSvi). It remains now only to verify (IDSi), (IDSvii) and (IDSix). Suppose now that (C_, A ... ) is not observable. Then by Lemma 12.2.2. in [BGR4l there is an z

e 0:".

not zero with C_(z - A... )-lz

we can solve for h+

e n.mXl(O') so that

= 0 identically in z.

From Theorem 4.5 (i) in [BR3I,

121

Ball and Rakowski

is in S,,(W, P) C S,,(w, P, Q) (here we use that (IDSvi) has already been verified). Furthermore, since ([ g~]

,A,,)

is observable by (NPDSi), and since C_(z -

A" )-lx = 0, by

Lemma 12.2.2. in [BGR4j again necessarily C+(z - A" )-l x is not identically zero. Then fin (4.6) is an element of S,,(w,P,Q) in violation of (ii-a). Hence we must have (C_,A".) observable, i.e. (IDSi) holds. Next suppose (IDSvii) is not true. Then we can find r E 1lLX1 such that Q+r ~ 1lmXl(0') but Q_r E 1lnXl(0"). Again by an argument based on Theorem 4.5 (i) in [BR3j, we can then produce an element h+ E 1lmXl(0") so that

[~~r]

E S,,(w, P)

c S,,(w, P, Q). The n

is an element of S,,(W, P, Q) in violation of (ii-a). Hence (IDSvii) must hold. Finally, suppose (IDSix) is violated. Then we can find x =F 0 in o;n. and r =F 0 in 1lLX1 so that L(z)

= C_(z -

A,,)-lx

+ Q_(z)r(z)

can produce h+ E 1lmXl(0') so that

[~~]

E

E

1lnXl(0'). From Theorem 4.5 (iii) in [BR3j we

S,,(w,P) C S,,(w,P,Q) where h_ = (I -

Also by Theorem 4.5 (i) from [BR3j there is 9 E 1lnx1 (0') so that

[g~] (z -

P~c)(Q_r).

A" )-1 X +

[g~)]

E

S,,(w, P) C S,,(w, P, Q) (here we use that (IDSvi) has already been verified). But then

+ Q_(z)r(z) is analytic on 0' and (NPDSix) is in force, we + Q+(z)r(z) is not analytic on 0". Hence f is an element of

However, since x =F 0, C_(z - A,,)-l x are guaranteed that C+(z - A,,)-I X

S,,(w, P, Q) in violation of (ii - a). Thus, (IDSix) must hold and Theorem 4.3 follows. The proof of Theorem 4.1 requires the following lemma, an adaptation of one of the main ideas in [Be]. LEMMA 4.4. Let 0' C 0;, let w as in (3.12) be a a-admissible interpolation data set, and let

122

Ball and Rakowski

be a block row rational matrix function with extended complete null-pole data set over u equal to

(w, [P+ P-1, [~~])

where

wis as in (4.1). Then a function W E nmxn is analytic on u and

satisfies the interpolation conditions (3.1 ')-(3.5') if and only if

[Wjz)] [021 02 2 0 231(n(m-K+n-L)X1(u) EB n LX1 )

C 0 ( n(m-K+n-L)x1(u) EB nLX1).

(4.7) PROOF:

As in the proof of Theorem 4.3, one can show that

[0 21 0 22 0 231( n(m-K+n-L)x1(u) EB n LX1 )

={C_(z -

A"y 1x: x E C n.} + nmX1(u) + Q_nLX1

S,,( (C_,A,.),(0,0),0),0,Q_).

=

Suppose W is analytic on u and satisfies the interpolation conditons (3.1 ')-(3.5'). By Lemma 4.4 in [BR51,

[lj] ({C_(Z-A,,)-1X:XEG:n.}+nmX1(u»)

C

S,,(w, P) C S,,(w, P, Q)

=

0(

n(m-K+n-L)x1(u) EB nLX1).

But by condition (3.5'),

and hence the inclusion (4.7) follows. Conversely, suppose that the inclusion (4.7) holds. Then

[lj] Q_nLX1

C

nr0(n(m-K+n-L)Xl(u)EBnLXl) ~ER

= 0(0 EB n LX1 ) = [8~] n LX1 and hence W satisfies (3.5'). Moreover, by an argument in the proof of Theorem 4.3 we know that

123

Ball and Rakowski

Using also that Q_r = 0 implies that Q+r = 0 (a consequence of (IDSvii», we see that the inclusion (4.7) implies that

[lj] c

S"

({C_(Z-A1r)-lx:x E o:n.}+1lnXl

(u»)

(w, [P+ p-1) = fm(m-K+n)xl(u).

Now Lemma 4.4 from [BR51 implies that W is analytic on u and satisfies the remaining interpolation conditions (3.1 ') - (3.4'). PROOF OF THEOREM

4.1. By Lemma 4.4 it suffices to show that W E 1lmxn satisfies

[lj]

[9 21 9 22 9 231( 1l(m-K+n-L)Xl(u)EIl1lLXl )

C 9 ( 1l(m-K+n-L)xl (u) EIl1lLXl)

(4.8)

if and only if W has a representation of the form

(4.9) where

Ql

E 1l(m-K)x(n-L)(u), Q2 E 1l(n-L)x(n-L)(u) are such that

[9 2l Ql

(4.10)

+ 9 22 Q2 9 231( 1l(n-L)xl(u) EIl1lLXl)

= [9 21 9 22 9 231(1l(m-K+n-L)Xl(u) Ell 1lLXl).

Suppose first the (4.9) and (4.10) hold. Rewrite (4.9) in the form

[lj]

[9 2l Ql

+ 9 22Q2

9 2319

[~~ ~].

Apply both sides to 1l(n-L)xl(u) EIl1l LXl and use (4.10) to get

[If] [9

21

9 22 9 231(1l(m-K+n-L)Xl(U) Ell 1lLXl)

= 9 [~~ ~]

(1l(n-L)Xl(U) EIl1lLXl).

Ball and Rakowski

124

Since Ql and Q2 are analytic on u, we have

and (4.8) follows. Conversely, suppose (4.8) holds. Note that r~r 'RLXl)

=

[lj]

9 23] ( 'R(m-K+n-L) Xl (u)E9

823'RLxl is a 'R-subspace of 'Rnxl of dimension L which (by (4.8)) is contained

n

r9('R(m-K+n-L)Xl(u) E9 'RLXl) == rER dimension count,

in

[lj] [821 822

[lj]

[~13] 'R LXl , also an 'R-subspace of dimension 8 23

823'RLXl ==

L. By

[~~:] 'RLXl.

Next note that the quotient module

[lj] [821 8 22 8 23]

('R(m-K+n-L)Xl(U) E9 'RLXl) /

[lj]

823'RLxl

is a free 'R(u)-submodule of

[qn-L,+ ] ql,+ ] with n - L generators; we may choose n - L generators of the form 8 [ qti- , .. ·8 qn-oL,- and

then set

Q+ == [ql,+ ... qn-L,+] E'R(m-K)x(n-L)(u), Q_ == [ql,- .. . qn-L,-] E'R(n-L)x(n-L)(u).

Then the left side of (4.8) has the representation

[lj] [92l 9 22 923] == 8

[~~ ~]

== [8 UQ1 9 21 Ql

('R(m-K+n-L)Xl(u) E9 'RLXl)

('R(n-L)Xl(U)E9'R LX1 )

l2 Q2 813] + + 9922Q2 9 23

('R(n-L)Xl(U) E9 'RLXl)

From equality of the bottom components in (4.11) we deduce that the pair (QhQ2) satisfies (4.10) and then (4.11) can be rewritten as

Ball and Rakowski

125

[lj] [921Ql + 9 22 Q2

923] ( n.(n-L)xl( u) Ell n. LX1 )

= [99 2111Q1 + 9 12 Q2 9 13 ] Ql + 8 22Q2 9 23

(4.12)

(n.(n-L)Xl(U) Ell n.LX1 )

Also from (4.10) we see that the square matrix function [9 21 Ql +9 22 Q2 9 23 ] cannot have determinant vanishing indenticaIly, and hence [9 21 Ql +9 22 Q2 9 23 ]-1 exists. Now (4.12) leads immediately to the representation (4.9) for W. The proof of Theorem 4.2 requires the following lemma. LEMMA 4.5. Suppose u,w and 9 =

[~11 ~12 ~13] 021 022 023

are as in the hypothesis of Theorem 4.2.

Then 9 satisfies

9(z)* J9(z) :5 i Ell 0, z E u (where J

= 1m Ell -In

and

i = Im-K Ell In-L)

if and only if [In-L 0][922 923r1921 has analytic

continuation to all of u. PROOF:

By assumption 9(z)*J9(z)

= i Ell -h for z E ou. In particular

[~t~] [912 9 13]- [~~~] [922 923] = -In' Hence

[9* ][922 9 23] 9~~

=

hn +

Z

E OU.

[9* ][912 9 13] for z E ou. As [922 9 23] is square, we conclude 9i~

that [922(Z) 9 23 (Z)] is invertible for Z E

ou.

Then for each fixed z we can rewrite the system of

equations

(4.13) (for z E ou) in the equivalent form

(4.14) where

= [912 9 13][922 9 23r1 U12 = 9 11 - [912 9 13][922 923r1921 g:~] = [822 823r1 U11

[

[g:] = -[822 823]-1821 .

126

Ball and Rakowski

Note that 0(z)* J0(z)

= j Ell -h on au is equivalent to

or equivalently, to

whenever z E au and (u,y,yO,v,w) satisfies (4.13). Since (4.13) and (4.14) are equivalent, we see that 0(z)* J0(z)

=j

EIl-h on

U12] au is equivalent to U(z) = [UU U2I U22 (z) being isometric on au.

Similarly, the validity of 0(z)* J0(z)

:s; j

hI

U32

Ell 0 on u is equivalent to

or otherwise stated, to

whenever z E u and (u,y,yO,v,w) satisfy (4.13). Since (4.13) and (4.14) are equivalent, we read off that this in turn is equivalent to [Im+n-L O]U(z) being contractive on u. In particular

must be contractive at all points of analyticity of u, and hence has analytic continuation to all of the set u. Conversely, to show that 0(z)* J0(z)

:s; jEllO on u is equivalent to showing that [Im+n-L 0]

U(z) has contractive values at all points of analyticity in u, or, equivalently by the maximum modulus theorem, that

have analytic continuation to all of u. By assumption we know only that

has such an analytic continuation. The fact that this forces Un, U12, and Un to have such an analytic continuation as well relies on the special structure of the extended null-pole subspace

e(1l(m-K+n-L)Xl(U) Ell 1lLXl) = s. . (w,[p+,P_J, [8~]) given by Theorem 4.3. Details for the I-block case (the case where K

= 0 and L = 0) are given in Theorem 13.2.3 in [BGR4Ji we leave

it to the reader to check that the general case here can be verified by analogous arguments.

Ball and Rakowski

PROOF OF THEOREM

127 4.2. Let a,w and 8 be as in the statement of Theorem 4.2. Suppose first

that 8(z)* J8(z) ::::: JED O(n-L)x(n-L) at all points z E a where 8(z) is analytic and that H E 1l(m-K)x(n-L)(a) with

IIHlloo < 1.

Then

(4.15) By Lemma 4.5 we know that [In-L 0][8 22 8 23 ]-18 21 is analytic on a and, from the proof of Lemma 4.5, that 11[8 21 823J-1821(z)1I invertible for all

Z

< 1 for z E Ba. Hence t[In-L 0][8 22 8 23 ]-18 21 H + In-L is

E Ba and for all t with 0 ::::: t ::::: 1. Hence wno det([In_L 0][8 22 823r1821H

+

In-L) = wno det In-L = 0, where wno X is the change of the argument of X(z) along Ba.

As we observed above that [In-L 0][8 22 823r1821H det([In_L 0][822 823r1821H

+ In-L)

+ In-L

is analytic on a, we conclude that

has no zero in a. From this we see that

From (4.15) we conclude that

[8 21 H

+ 8 22 8 23] (1l(n-L)Xl(a) ED1lL)

= [8 22 8 23 ] ( 1l(n-L)xl( a) ED1lL ).

(4.16) On the other hand,

[8 21 8 22 8 23 ] ( 1l(m-K+n-L)X1(a) ED1lL) ( 4.17)

= [8 22 8 23 ][[8 22 823J-1821 In] (1l(m-K+n-L)X1(a)ED1l L ).

But by Lemma 4.5 [In-L 0][8 22 823]-1821 is analytic on a, so

[[822823]-1821 In] ( 1l(m-K+n-L)x1 (a) ED1lLX1) (4.18)

= 1l(n-L)Xl(a) ED1lLX1.

Combining (4.17) and (4.18), we obtain

Ball and Rakowski

128

[9 21 9 22 9 231 ( R(m-K+n-L)xl(U) E9 RLX1) = [9 22 9 231 ( R(n-L)xl(U) E9 RLX1).

(4.19)

Combine (4.19) and (4.16) to see that the pair (H,I) is an admissible parameter pair in the sense of Lemma 4.4. Hence, by Lemma 4.4 we know that W = [9 11 H + 9 12 9 13 )[9 21 H + 9 22 923r 1 is analytic on u and satisfies the interpolation conditions (3.1') - (3.5'). Moreover, for z E 8u,

where 'I/J

= [9 21 H + 9 22

923r 1. Hence IIW(z)1I

< 1 for z E

8u. By the maximum modulus

principle, IIW(z)1I < 1 for z E u. Conversely, suppose that there exists a function W E Rmxn which is analytic on u, satisfies the interpolation conditions (3.1') - (3.5') and has IIWlloo < 1. Then by Theorem 4.1 we know that there are Ql E R(m-K)x(n-L)(u) and Q2 E R(n-L)x(n-L)(u) such that

[9 21 Ql (4.20)

+ 9 22 Q2

9 231( R(n-L)xl(u) E9 RLX1)

= [9 21 9 22 9 231(R(m-K+n-L)Xl(u) E9 RLX1),

from which we recover W as

(4.21) An alternative way to write (4.21) is

= [9 21 Ql + 9 22 Q2

9 231. Since IIWlloo < 1 and on 8u by assumption, we have, for z E u, where 'I/J

e satisfies condition (b) in Theorem 4.2

129

Ball and Rakowski

( 4.22) Thus Q2(Z) cannot have any null space for z E

0',

and hence, as it is square, Q2(Z)-1 exists for

z E au. Also (4.22) then implies that H(z) = Ql(Z)Q2(Z)-1 iswell defined and satisfies IIH(z)1I < 1 for z E

au. Next note from (4.20) the inclusion

[0 22 0 23 ]( 'R("-L)Xl(u)ED'R LX1 )

c [0 21 Ql + 0 22 Q2 023] ( 'R(,,-L)Xl(u) ED 'RLXl).

(4.23)

We also know from the proof of Lemma 4.5 that [0 22 0 23 ] is invertible on

au.

Then (4.23) can be

written in the form

'R(n-L)Xl(U) ED'RLX1 (4.24)

C ([0 22 023r1021[Ql 0]+

[~2 ~])('R(n-L)Xl(u)ED'RLXl).

A

= [1,,-L

0][022 023r1021Ql + Q2 has size (n -

has size L X (n - L). In this notation (4.24) assumes the form 'R(n-L)Xl(U) ED 'R LX1 C This in turn requires

(4.25) Note that

JJ

[~2 ~] has the form [~ where L) X (n - L) and B = [01£][022 023r1021Ql

Note that the nxn matrix function [022 023r 1021[Ql 0]+

[~ ~]

('R(n-L)Xl(u) ED 'RLXl).

130

Ball and Rakowski

A = ([In-L 0][0 22 023r 10 21 H proof of Lemma 4.5) and where IIH(z)1I

+ I)Q2

where 11[0 22 0 23r 1 0 21 11 < Ion au (from the

< 1 and Q2(Z) is invertible on au from observations made

above. Hence A(z) is invertible on az and wno det A(z) is well defined. The inclusion (4.25) gives us that

On the other hand,

(4.27) As was observed above, II[In-L 0][0 22 023r1021HII

argument wno det ([In-L 0][0 22 023r1021H

+ I)




(ii) together with the parametrization of stabilizing compensators (1) and

of the associated closed loop transfer functions Tzw (2). Let K be a given compensator. As explained in Section 5.1, K stabilizes P if and only if one can solve the system of equations (5.7) uniquely for stable z, U, YI in terms of any prespecified stable w, VI. V2. In more concrete form, this means: given any stable hI. h2' h3 there must exist kl. k2' k3, k .. with kl. k2' k3 stable and unique such that

(5.9)

[

~I

]

h2 - K h3

Note that

[~2J

= [kl_p~k2k: P~~3k: p~~:"l k2 - K k3

P;2 P03] is square, so in fact, necessarily the pair (hI. h2 - K h3) must determine

k2' k3, k .. uniquely.

Now suppose that K with coprime factorization K = N K Dil is a stabilizing compensator for P. Then in particular we can solve (5.9) with h2 we get that necessarily k2

= 0, h3 = 0 and hI arbitrary.

= NKg, k3 = DKg for some stable g.

From k2 - K k3

=0

To simultaneously solve the second

equation in (5.9), one must have such a stable 9 together with a k .. (not necessarily stable) so that

-hI = (P2I N K

+ P 22 DK )g + P 23k ...

From the first equation in (5.9), in addition we must have (PUNK

+ PI2DK)9 + P 23k .. =: ki

E 1?mXI(u).

As hI is an arbitrary element of 1?nXI(u), we have thus verified condition (ii-b) in the statement of the theorem. To verify (ii-a), suppose that k2 E 1?(m-K)xI(u), k3 E 1?(n-L)xl(u), k.. E 1?LXI are such that

Ball and Rakowski

139

Then the system of equations

o = - 1'21 k2 -

1'22k3 - 1'23k4

k2 - Kk3 = k2 - Kk3

uniquely detennines k2 and k3, since well-posedness implies the regularity of the matrix function

[~2}

K

1' 2 1'03 ]. Since K is stabilizing for P, from the first equation in (5.9) we see that neces-

sarily

This verifies (ii-a). Next we verify

[1'21 NK

+ 1'22 DK P231(n(n-L)x1(u) $nLX1 )

= [1'21 1'22

(5.10)

1'231 ( n(m-K+n-L)x1 (u) $

nLX1).

To see this, choose h1 = 0 and let h2 and h3 be general stable functions in (5.9). From the last equation in (5.9), there must be agE n(n-L)x1(u) with k2

= h2 + NK9,k3 = h3 + DKg.

Now the

second equation in (5.9) says there is some choice of such a 9 and a k4 E n LX1 such that

This verfies (5.10). Conversely, suppose S satisfies (ii-a) and (ii-b) and K E n(m-K)x(n-L) has a coprime factorization K = NKDi/ such that the pair (NK' DK) satisfies (5.10). To show that K is stabilizing for P we need to verify that (5.9) is solvable for k1 E nmX1(u), k2 E n(m-K)x1(u), k3 E n(n-L)x1(u) and k4 E n LX1 for any h1 E nnX1(u), h2 E n(m-K)x1(u) and h3 E n(n-L)x1(u).

By linearity it suffices to consider two special cases.

= 0,h3 = O. By (ii-b) there are stable k2,k3 and a not necessarily stable k4 so that -h1 = 1'21 k2 +1'22k3 +1'23 k4 and such that k1 := 1'llk2+1'12k3 +1'13k4 Case 1: h1 E nnx1(u),h2

is stable. Then (kt, k2' k3, k 4) is the desired solution of (5.9). Case 2: h1 = 0,h 2 E n(m-K)x1(u),h3 E n(n-L)x1(u). By (5.10) we can find 9 E 1l(n-L)x1(u) and k4 E 1lLX1 so that

140

Ball and Rakowski

-[,P21h2 + p22h3] = (p21 N K + p22 DK)9 + p 23k4. Then k2

= h2 + NKg, k3 = h3 + DK9, k4 give a solution of the last two equations in (5.9).

From

(ii-b) combined with (ii-a) we see that also kl := p ll k 2+p ll k 2+p23k4 must be stable as well. Then

(khk2'k3'k4) gives the desired solution of (5.9) in this case. Hence K = NKDi/ is stabilizing for'P as asserted. To complete the proof of Theorem 5.1 it remains only to verify (2). But this follows from the characterization (1) of stabilizing compensators and the characterization of the range of the linear fractional map (NK' DK)

->

[pllNK

+ p12DK, p 13][p21 NK + p 22 DK, p23r 1 for stable pairs

(NK' DK) satisfying (5.10) given by Theorem 4.1. The side condition that [0 In-L O]p- L

[lj]

\If

[In-L 0] be biproper is imposed to restrict (NK' DK) to pairs for which K = NKDi/ is well defined and proper. References [ABDS] D. Alpay, P. Bruinsma, A. Dijksma, H.S.V. de Snoo, Interpolation problems, extensions of symmetric operators and reproducing kernel spaces I, in Topics in Matrix and Operator

Theory (ed. by H. Bart, I. Gohberg and M. A. Kaashoek), pp. 35-82, OT 50, Birkhiiuser Verlag, Basel-Boston-Berlin, 1991. [B] V. Belevitch, Classical Network Theory, Holden Day, San Francisco, 1968. [BC] J. A. Ball and N. Cohen, Sensitivity minimization in an HOO norm: parametrization of

all suboptimal solutions, Int. J. Control 46 (1987), 785-816. [BCRR] J. A. Ball, N. Cohen, M. Rakowski and L. Rodman, Spectral data and minimal divisibility of nonregular meromorphic matrix functions, Technical Report 91.04, College of William &;

Mary, 1991.

[BGK] H. Bart, I. Gohberg and M. A. Kaashoek, Minimal Factorization of Matrix and Operator

Functions, Birkhauser, 1979 [BGR1] J. A. Ball, I. Gohberg and L. Rodman, Realization and interpolation of rational matrix Functions, in Topics in Interpolation Theory of Rational Matrix Functions (ed. I. Gohberg), pp. 1-72, OT 33, Birkhauser Verlag, Basel Boston Berlin, 1988. [BGR2] J. A. Ball, I. Gohberg and L. Rodman, Two-sided Lagrange-Sylvester interpolation problems for rational matrix functions, in Proceeding Symposia in Pure Mathematics, Vol. 51, (ed. W. B. Arveson and R. G. Douglas), pp. 17-83, Amer. Math. Soc., Providence, 1990. [BGR3] J. A. Ball, I. Gohberg and L. Rodman, Minimal factorization of meromorphic matrix functions in terms oflocal data, Integral Equations and Operator Theory, 10 (1987),309348. [BGR4] J. A. Ball, I. Gohberg and L. Rodman, Interpolation

0/ Rational Matrix Functions, OT

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45, Birkhii.user Verlag, Basel-Boston-Berlin, 1990. [BGR5] J. A. Ball, I. Gohberg and L. Rodman, Sensitivity minimization and tangential NevanlinnaPick interpolation in contour integral form, in Signal Processing Part II: Control Theory

and Applications (ed. F. A. Griinbaum et al), IMA Vol. in Math. and Appl. vol. 23, pp. 3-25, Springer-Verlag, New York, 1990. [BGR6] J. A. Ball, I. Gohberg and L. Rodman, Tangential interpolation problems for rational matrix functions, in Proceedings of Symposium in Applied Mathematics vol. 40, pp. 5986, Amer. Math. Soc., Providence, 1990. [BH] J. A. Ball and J. W. Helton, A Beurling-Lax theorem for the Lie group U(m, n) which contains most classical interpolation, J. Operator Theory, 9, 1983, 107-142. [BHV] J. A. Ball, J. W. Helton and M. Verma, A factorization principle for stabilization of linear control systems, Int. J. of Robust and Nonlinear Control, to appear. [BR1] J. A. Ball and M. Rakowski, Minimal McMillan degree rational matrix functions with prescribed zero-pole structure, Linear Algebra and its Applications, 137/138 (1990), 325349. [BR2] J. A. Ball and M. Rakowski, Zero-pole structure of nonregular rational matrix functions, in Extension and Interpolation of Linear Operators and Matrix Functions (ed. by I. Gohberg), OT 47, pp. 137-193, Birkhii.user Verlag, Basel Boston Berlin. [BR3] J. A. Ball and M. Rakowski, Null-pole subspaces of rectangular rational matrix functions,

Linear Algebra and its Applications, 159 (1991),81-120. [BR4] J. A. Ball and M. Rakowski, Transfer functions with a given local zero pole structure, in

New Trends in Systems Theory, (ed. G. Conte, A. M. Perdon and B. Wyman), pp. 81-88, Birkhii.user Verlag, Basel-Boston-Berlin, 1991. [BR5] J. A. Ball and M. Rakowski, Interpolation by rational matrix functions and stability of feedback systems: the 2-block case, preprint. [BRan] J. A. Ball and A. C. M. Ran, Local inverse spectral problems for rational matrix functions,

Integral Equations and Operator Theory, 10 (1987), 349-415. [CP] G. Conte, A. M. Perdon, On the causal factorization problem, IEEE Transactions on

Automatic Control, AC-30 (1985),811-813. [D] H. Dym, J. contractive matrix functions, interpolation and displacement rank, Regional conference series in mathematics, 71, Amer. Math. Soc., Providence, R.I., 1989. [DGKF] J. C. Doyle, K. Glover, P. P. Khargonekar and B. A. Francis, State-space solutions to standard H2 and HCO control problems, IEEE Trans. Auto. Control, AC-34, (1989), 831-847. [F] G. D. Forney, Jr., Minimal bases ofrational vector spaces, with applications to multivariable linear systems, SIAM Journal 0/ Control, 13 (1975),493-520.

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[FF] C. Foias and A. E. Frazho, The Commutant Lifting Approach to Interpolation Problems, Birkhii.user Verlag, Basel-Boston-Berlin, 1990. [Fr] B. A. Francis, A Course in Hoo Control Theory, Springer Verlag, New York, 1987. [GK] I. Gohberg and M. A. Kaashoek, An inverse spectral problem for rational matrix functions and minimal divisibility, Integral Equations and Operator Theory, 10 (1987),437-465. [Hu] Y. S. Hung, Hoo interpolation ofrational matrices, Int. J. Control, 48 (1988), 1659-1713. [K] T. Kailath, Linear Systems, Prentice-Hall, Englewood Cliffs, N. J., 1980.

[Ki] H. Kimura, Directional interpolation approach to Hoo-optimization and robust stabilization, IEEE

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Auto. Control, AC-32 (1987), 1085-1093.

[M] A. F. Monna, Analyse non-archimedienne, Springer, Verlag, Berlin Heidelberg New York, 1970. [McFG] D. C. McFarlane and K. Glover, RobtlSt Controller Design Using Normalized Coprime Factor Plant Descriptions, Lecture Notes in Control and Information Sciences Vol. 138, Springer-Verlag, New York, 1990. . [NF] B. Sz.-Nagy and C. Foias, Harmonic Analysis of Operators on Hilbert Space, American Elsevier, New York, 1970.

[R] M. Rakowski, Generalized Pseudoinverses of Matrix Valued Functions, Int. Equations and Operator Theory, 14 (1991),564-585. [YBL] D. C. Youla, J. Bongiorno and Y. Lu, Single loop stabilization of linear multivariable dynamic plants, Automatica, 10 (1974), 151-173. [YJB] D. C. Youla, H. A. Jabr and J. J. Bongiorno, Modem Wiener-Hopf design of optimal controllers: I and II, IEEE

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Auto. Control, AC-291 (1977),3-13.

[VJ M. Vidyasager, Control Systems Synthesis: A Factorization Approach, MIT Press, Cambridge, Mass., 1985.

Department of Mathematics

Department of Mathematics

Vn-ginia Tech

Southwestern Oklahoma State University

Blacksburg, VA 24061

Weatherford, OK 73096

MSC: Primary 47A57, Secondary 93B52, 93B36

Operator Theory: Advances and Applications, Vol. 59 © 1992 Birkhauser Verlag Basel

143

MATRICIAL COUPLING AND EQUIVALENCE AFTER EXTENSION H. Bart and V.E. Tsekanovskii

The purpose of this paper is to clarify the notions of matricial coupling and equivalence after extension. Matricial coupling and equivalence after extension are relationships that mayor may not exist between bounded linear operators. It is known that matricial coupling implies equivalence after extension. The starting point here is the observation that the converse is also true: Matricial coupling and equivalence after extension amount to the same. For special cases (such as, for instance, Fredholm operators) necessary and sufficient conditions for matricial coupling are given in terms of null spaces and ranges. For matrices, the issue of matricial coupling is considered as a completion problem.

1

Introduction

Let T and S be bounded linear operators acting between (complex) Banach spaces. We say that T and S are matricially coupled if they can be embedded into 2 x 2 operator matrices that are each others inverse in the following way

(1) . This notion was introduced and employed in [7]. In the addendum to [7], connections with earlier work by A. Devinatz and M. Shinbrot [18] and by S. Levin [38] are explained (d. [46]). For a recent account on matricial coupling, see the monograph [19]. Concrete examples of matricial coupling, involving important classes of operators, can be found in [2], [7], [8], [10], [19], [21], [24], [26], [33] and [45]. The operators T and S are called equivalent after extension if there exist Banach spaces Z and W such that T EB Iz and S EB Iw are equivalent operators. This means that there exist invertible bounded linear operators E and F such that

(2)

Bart and 1Sekanovskii

144

Two basic references in this context are [22] and [23]. The general background of these papers is the study of analytic operator functions. Thus the relevant issue in [22] and [23] is analytic equivalence after extension, i.e., the situation where the operators T, S, E and F in (2) depend analytically on a complex parameter. Other early references with the same background are [14], [16], [30], [31], [32], [34], [41] and [42]. For a recent application of analytic equivalence after extension involving unbounded operators, see [35]. Ordinary analytic equivalence (without extension) plays a prominent role in [1], [29] and [37]. More references, also to publications not dealing with operator functions but with single operators, will be given in Section 3. Evidently, operators that are equivalent after extension have many features in common. Although this is less obvious, the same conclusion holds for operators that are matricially coupled. The reason behind this is that matricial coupling implies equivalence after extension. For details see [7] and [19], Section IlIA. The main point of the present paper is the observation that not only does matricial coupling imply equivalence after extension, in fact the two concepts amount to the same. The proof involves the construction of a coupling relation (1) out of an equivalence relation of the type (2). This is the main issue in Section 2. Section 3 contains examples. Two examples are directly taken from the literature; in the other three, known material is brought into the context of matricial coupling. Along the way, we give additional references. In Section 4, we specialize to generalized invertible operators. For such operators, matricial coupling is characterized in terms of null spaces and ranges. An example is given to show that the invertibility condition is essential. Things are further worked out for finite rank operators, Fredholm operators and matrices. For matrices, one has the following simple result: IT T is an mT x nT matrix and S is an ms x ns matrix, then T and S are matricially coupled if and only if

rankT - rankS = mT - ms = nT - ns.

(3)

Section 4 ends with a discussion of matricial coupling of matrices viewed as a completion problem: Under the assumption that (3) is satisfied, construct matrices To, TI, T2 , So, SI and S2 of the appropriate sizes such that (1) is fulfilled. Extra details are provided for the case when T and S are selfadjoint. A few remarks about notation and terminology. The letters 'R and C stand for the real line and the complex plane, respectively. All linear spaces are assumed to be complex. The identity operator on a linear space Z is denoted by Iz, or simply I. By dimZ we mean the dimension of Z. For two Banach spaces X and Y, the notation X ~ Y is used to indicate that X and Yare isomorphic. This means that there exists an invertible bounded linear operator from X onto Y. IT X is a Banach space and M is a closed subspace of X, then X / M stands for the quotient space of X over M. The dimension of X / M is called the codimension of M (in X) and written as codimM. The null space and range of a linear operator T are denoted by ker T and im T, respectively. The symbol EB signals the operation of taking direct sums, not only of linear spaces, but also of operators and matrices.

Acknowledgement. The foundation for this paper was laid in May 1990 while the first author was visiting Donetsk (USSR) on the invitation of E.R. Tsekanovskii, the father of the second author.

Bart and Thekanovskii

2

145

Coupling versus equivalence

We begin by recalling the notion of matricial coupling (d. [7] and [19]). Let Xl! X 2 , YI and l'2 be (complex) Banach spaces. Two bounded linear operators T: Xl ~ X 2 and S: Yi ~ Y2 are said to be matricially coupled if they can be embedded into invertible 2 x 2 operator matrices

(4) (5) involving bounded linear operators only, such that

(6) The identity (6) is then called a coupling relation for T and S, while the 2 x 2 operator matrices appearing in (4) and (5) are referred to as coupling matrices. Next, let us formally give the definitions of equivalence and equivalence after extension. The operators T : Xl ~ X 2 and S : Yi ~ l'2 are called equivalent, written T", S, if there exist invertible bounded linear operators Vi : Xl ~ Yi and ~ : l'2 ~ X 2 such that T = ~SVi. Generalizing this concept, we say that T and S are equivalent after extension if there exist Banach spaces Z and W such that T E9 Iz '" S E9 Iw. In this context, the spaces Z and Ware sometimes referred to as extension spaces. Ordinary equivalence, of course, corresponds to the situation where these extension spaces can be chosen to be the trivial space. From [7] it is known that matricial coupling implies equivalence after extension (d. [19], Section IlIA). Our main result here is that the converse is also true. Theorem 1 Let T : Xl ~ X 2 and S : Yi ~ l'2 be bounded linear operators acting between Banach spaces. Then T and S are matricially coupled if and only if T and S are equivalent after extension. Proof. Assume T and S are equivalent after extension, i.e., there exist Banach spaces Z and W such that T E9 Iz and S E9 Iw are equivalent. Let

and F =

(~~: ~~:): Xl E9 Z ~ Yi E9 W

be invertible bounded linear operators such that T E9 Iz = E(S E9 Iw)F, i.e.,

(En E12) ( TO) o Iz = E21 E22

(S 0 ) (Fn F12) 0

Iw

F21 F22

.

Bart and Thekanovskii

146 Write the inverses E- 1 and F- 1 as E- 1 = (

E(-l) E (-1) ) 12 n : X 2 E9 Z - E (-1) E (-1) 21

1'2 E9 W

22

and 1 F- = (

(-l) F 11

F(-1») 12

F (-l)

F(-l)

21

: Yi E9 W - - Xl E9 Z.

22

A straightforward computation, taking into account the identities implied by the above set up, shows that the operators

( T -En)

: Xl E9 1'2

F11 F12 E 21

and (

-- X 2 E9 Yi

",(-1»)

(-1)E(-l) F12 21 .l"n 8 E (-1) -

: X 2 E9 Yi -- Xl E9 1'2

11

are invertible and each others inverse. Thus (

T -Ell F11 F12E21

(F(-1) E(-1) 1Z 21

-1 )

=

_ Ef~l)

is a coupling relation for T and 8. This proves the if part of the theorem. For the proof of the if part, we could simply refer to [7] or [19]. For reasons of completeness and for later reference, we prefer however to give a brief indication of the argument. Suppose T and 8 are matricially coupled with coupling relation (6). Following [7], we introduce

F =

(~1 ~~): Xl E9 1'2 -- Yl E9 X

2•

Then E and F are invertible with inverses E- 1 =

(~!2

8:£0): 2 1'2 __ 1'2 X E9

E9 X 2 ,

Bart and 1Sekanovskii

147

A direct computation shows that T E9 I Y2 = E(S E9 I x2 )F. Thus T E9 IY2 and S E9 IX2 are equivalent. This completes the proof. Of particular interest is the case when the operators T and S depend analytically on a complex parameter. Theorem 1 and its proof then lead to the conclusion that analytic matricial coupling amounts to the same as analytic equivalence after extension (d. [7], Section 1.1 and [19], Section lIlA). Another remark that can be made on the basis of the proof of Theorem 1 is the following. Suppose T : Xl ---+ X 2 and S: Yi ---+ l'2 are equivalent after extension, i.e., there exist Banach spaces Z and W such that T E9 Iz '" S E9 Iw. Then Z and W can be taken to be equal to l'2 and X 2 , respectively. Another possible choice is Z = Yi and W = Xl (d. [7], Section 1.1). Thus, if the underlying spaces Xt,X2 , Yi and l'2 belong to a certain class of Banach spaces (for instance separable Hilbert spaces), then the extension spaces Z and W can be taken in the same class. Roughly speaking, equivalence by extension, if at all possible, ·can always be achieved with "relatively small" or "relatively nice" extension spaces. We conclude this section with some additional observations. But first we introduce a convenient notation. Let T : Xl ---+ X 2 and S : Yi ---+ l'2 be bounded linear operators acting between Banach spaces. We shall write T ~ S when T and S are matricially coupled or, what amounts to the same, T and S are equivalent after extension. The relation ~ is reflexive, symmetric and transitive. This is obvious from the viewpoint of equivalence after extension. In terms of matricial coupling things are as follows. Reflexivity is seen from (

T

IXl

-Ix2)

-1

= (

o

0

Symmetry is evident from the fact that (6) can be rewritten as

Finally, if T ~ S and S ~ R, with coupling relations

then T ~ R with coupling relation

This can be verified by calculation. The relation ~ implies certain connections between the operators involved. Those that are most relevant in the present context are stated in the next proposition.

Bart and 1Sekanovskii

148

Proposition 1 Let T : Xl --+ X 2 and S : Yi --+ Y2 be bounded linear operators, and assume T :.., S. Then ker T ~ ker S. Also im T is closed if and only if im S is closed, and in that case Xdim T ~ Y2/im S. All elements needed to establish the proposition can be found in [7], Section 1.2 and [19], Section 111.4. The details are easy to fill in and therefore omitted. We take the opportunity here to point out that there is a misprint in [7]. On the first line of [7], page 44, the symbol B21 should be replaced by A 12 •

3

Examples

Interesting instances of matricial coupling can be found in the publications mentioned in the Introduction. These concern integral operators on a finite interval with semi-separable kernel, singular integral equations, Wiener-Hopf integral operators, block Toeplitz equations, etc .. Here we present five examples. In the first three, known material is brought into the context of matricial coupling. The fourth example can be seen as a special case of the Example given in [7], Section 1.1, and the fifth summarizes the results of [7], Section IV.l and [19], Section XIII.8. Example 1 Suppose we have two scalar polynomials

The resultant (or Sylvester matrix) associated with a and b is the 2m x 2m matrix ao

0

R = R(a, b) =

a1 ao

0

a m -1 a m -2

0

be

~

0

be

0

1 a m -1

ao

0 0 a m -1

1 bm - 1 bm - 2 bm - 1

0

0 1

1

1

be

bm - 1 1

The Bezoutian (or Bezout matrix) associated with a and b is the m x m matrix B = B(a, b) = (bi ;)ij=l

given by a(.\)b(l') - a(l')b(.\) _ ~ b.. .\i-1 ;-1 .\_

I'

-~

iJ=1

~

I'

.

149

Bart and 1Sekanovskii

As is well-known, the matrices Rand B provide information about the common zeros of a and b (see, e.g., [36], Section 13.3). Our aim here is to show that R and B are matricially coupled. Matrices are identified with linear operators in the usual way. It is convenient to introduce the following auxiliary m x m matrices:

al a2

a2 a3

~-1

1

1

0

o

0

S(a) = am -l

1

1

0

ao al 0 ao

am -2 am -l ~-3 am -2

T(a) = 0 0

0 0

h[~

ao 0 0 0

0 1 1 0

1 0

0 0 0

al ao

Observe that R = R( a, b) can be written as

T(a) JS(a)) R = ( T(b) JS(b) .

(7)

From [36], Section 13.3 we know that

S(a)T(b) - S(b)T(a)

= B,

S(a)JS(b) - S(b)JS(a) = O.

Clearly J2 = 1m, where 1m stands for the m x m identity matrix. A simple calculation now shows that 0 ) T(a) JS(a) ( T(b) JS(b) -S(a)-1 1m

0

0

-1

=

(0

0

1m)

S(a)-IJ 0 -S(a)-IJT(a) S(b) -S(a) B

.

In view of (7), this is a coupling relation for Rand B. By the results of Section 2, we have Rffilm ,.... Bffil2m , i.e. Rand B are equivalent by two-sided extension involving the m x m and 2m x 2m identity matrix. In the present situation things can actually be done with one-sided extension. In fact, R,.... B ffi 1m. Details can be found in [36], Section 13.3 (see also the discussion on finite rank operators in Section 4 below).

Bart and lSekanovskii

150

The equivalence after extension of the Bezoutian and the resultant already appears in [20]. For an analogous result for matrix polynomials, see [39]. It is also known that the Vandermonde matrix and the resultant for two matrix polynomials are equivalent after extension (cf. [25]). Example 2 This example is inspired by [40], Section 3. Let A : Z W, B : X Z, C :W Y, and D : X Y be bounded linear operators acting between Banach spaces. Then D + CAB is a well-defined bounded linear operator from X into Y. Put M = (

Then D

+ CAB,!, M (

-Iw C

o

0 A ) D O : W ffi X ffi Z B -Iz

W ffi Y ffi Z.

and the identity

D+CAB -C -Iy -CA AB -Iw 0 -A o Ix 0 0 BOO -Iz

)_1

o -Iw C

o

is a coupling relation for D + CAB and M. This coupling relation implies that

D

+ CAB ffi IWEllYez '" M

ffi I y ,

+ CAB ffi Iwexez '" M

D

ffi Ix.

Both equivalences involve two-sided extensions, but, as in Example 1, things can be done with one-sided extension. Indeed, it is not difficult to prove that M is equivalent to D + CAB ffi Iwez. The details are left to the reader. Example 3 Consider the monic operator polynomial L(~) = ~n I

+ ~n-l A n - 1 + ... + ~Al + Ao.

Here Ao, ..• , An-I are bounded linear operators acting on a Banach space X. Put 0

I

0

0

I

0 0

0

0

CL =

0 0 -Ao -AI

0 -A2

:xn -X".

0

0 I -An- 1

It is well-known that L(~)

ffi Ix n-l

'"

M - CL

151

Bart and 1Sekanovskii

and, in fact, we have a case here of analytic equivalence after one-sided extension. Clearly it is also a case of linearization by (one-sided) extension. For details, see [3], [27], [28], [36] and [44]. Now, let us consider things from the point of view of matricial coupling. For k = 0, ... , n - 1, put

so L o(>') = -I in particular. Then we have the coupling relation L(>.) I >.I >.21

L n- 1(>.) L n- 2(>.) 0 0 -I 0 ->.I -I

_>.n-l I

_>.n-2 I 0 0 0

L2(>') Ll(>') Lo(>') 0 0 0 0 0 o. 0 0 0

_>.n- 31

I 0 >.I -I 0 >.I

0 0 0 -I Ao Al

->.I

-I

0 0 0

0 0 0

0 0 0

0

>.I

-I

-1

=

0

An-3 A n - 2 >.I + A n - l

showing that L(>.) and >.I - CL are matricially coupled. Note that this is a case of analytic matricial coupling. Example 4 Let A : X --+ X, B : Y --+ X, C : X --+ Y and D : Y --+ Y be bounded linear operators between Banach spaces. For>. in the resolvent set peA) of A, we put W(>') = D

+ C(>'lx -

A)-l B.

(8)

Assume that D is invertible and write AX = A - BD-IC. It is well-known that

(9) and, in fact, we have another case here of analytic equivalence after (two-sided) extension. For details and additional information, see [4], Section 2.4. Considering things from the viewpoint of matricial coupling, we see that W(>') ~ >.Ix - AX with coupling relation (

W(>.) - (>.Ix - A)-1 B

-C(>.Ix - A)-I) -1 (>.lx - A)-1

(D-I =

D- 1C )

BD-l >.Ix - AX

Note that this is again a case of analytic matricial coupling.

Bart and Thekanovskii

152

An expression of the type (8) is called a realization for W. Under very mild conditions analytic operator functions admit such realizations. For instance, if the operator function W is analytic on a neighbourhood of 00, then W admits a realization. For more information on this issue, see [4]. Whenever an operator function W can be written in the form (8) with invertible D, it admits a linearization by two-sided extension (9), and hence certain features of it can be studied by using spectral theoretical tools. Under additional (invertibility) conditions on B or e, even linearization by one-sided extension, i.e. analytic equivalence of the type

W(A) ffi I

rv

AIx - AX,

can be achieved (cf. [4], Section 2.4; see also [31]).

Example 5 Let K : Lp([O, 00), Y) given by

--+

[K== r.p(z,,I) + ,,(z, y).

Put

Fujii

163

Then we have a Hilbert space 1l as the completion of VIN and the derivative. A, B on

1l by

< Ai,ii >= ~(z,y) Since A and B commutes by A

+B

and

< Bi, ii >= ,p( z, y).

= I, we can define a positive sesquilinear form

../rl

by

Then they showed that its definition does not depend on representations: THEOREM(Pusz-Woronowicz). If there ezi." a map,

21-+

i, onto a den.e

.et of a Hilbert .pace H with commuting derivative. C and D

then

More generally, if f(t,.) is a suitable (homogeneous) function (see also [24]), then one can define f(~, ,p) by f(~,,p)(z,y)

=< f(C,D)i I Y>H·

3. OPERATOR MEANS AND OPERATOR MONOTONE FUNCTIONS Seeing these objects, Ando [21 introduced some operator means of positive operators on a Hilbert space:

geometric mean:

AgB == max {X

harmonic mean :

AhB == max { X

As a matter of fact, we have AhB

= 2A: B

and

!) ~ o}, ~ 0I (2: 2~) ~ (i i)}· ~ 0 I (i

< AgBz,y >= v'< A·,· >< B·,· >(2,y).

Like numerical case, the following inequalities hold:

A+B AhB::; AgB::; AaB == - 2 - .

164

Fujii

In their common properties, note the following inequality called the tran.former one:

If T is invertible then the equality holds in the above. Indeed, we have

T*(AmB)T ~ T· ATmT* BT = T*T·- 1 (T· ATmT· BT)T- 1 T ~ T*(AmB)T. In particular,

for invertible A and B. On the other hand, Lowner [21] defined the notion of operator monotone functions. A real-valued (continuous) function

I

on an interval I in the real line is called

operator monotone on I and denoted by f E OM(I) if

A

~

B implies I(A)

~

c

I.

for all selfadjoint operators A, B with o-(A), o-(B)

Then, for an open interval I, a function

I(B)

I

is monotone-increasing analytic

function and characterized in some ways: 1. Every LO'fl1Rer matriz is positive semi-definite:

1(';») ~ 0

( /(t i ) ti -';

for

'1 < tl 0

for 1m z > O.

3. I has a suitable integral representation, see also [2,5]. Now we see the general operator means due to Kubo and Ando [20]. A binary operation m among positive operators on a Hilbert space is called an operator mean if it satisfies the following four axioms:

monotonousness: lower continuity:

A

~

0, B

~

D ===> AmB

~

OmD,

A,. ! A, B.. ! B ===> A,.mB. ! AmB,

165

Fujii

T·(AmB)T:5

transformer inequality:

r

ATmT· BT,

and

AmA=A.

normalization:

A nonnormalized operator mean is called a connedion. For invertible A, we have

(1) and fm(z)

=

1mz is operator monotone on [0,00). (Note that fm(z) is a scalar since

fm(z) commutes with all unitary operators by the transformer 'equality'.) By making use of an integral representation of operator monotone functions, we have a positive Radon measure I'm on [O,ooJ with

AmB = aA + bB +

(2) where a

=

fm(O)

= Pm({O})

and b

1

l+t (tA) : B-dpm(t) (0,00) t

= inftfm(l/t) = Pm({OO}).

So the heart of the

Kubo-Ando theory might be the following isomorphisms among them:

THEOREM (Kubo-Ando).

Map. m

f-+

fm and m

f-+

Pm defined by (1)

and (2) give affine order-i.omorphi.m. from the connection. to the nonnegative continuou. operator monotone function. on [0,00) and the po.itive Radon mea.ure. on [O,ooJ. If m

u

an operator mean, then fm(l) = 1 and P i. a probability mea.ure. Here fm (resp. Pm) is called the rep relenting fundion (resp. mea.ure) for m.

4.

OPERATOR CONCAVE FUNCTIONS AND JENSEN'S IN-

EQUALITY Like operator monotone functions, a real-valued (continuous) function F on I is called operator concave on I and denoted by F E OC(I) if

F(tA + (1- t)B)

~

tF(A) + (1 - t)F(B)

for all selfadjoint operators A, B with (T(A), (T(B)

c I.

(0 :5 t :5 1)

(ll -F is operator concave, then F

is called operator cORvell.) Then, for an open interval I, a function F is concave analytic

166

Fujii

function and characterized by (see [5]) F[a)(z) == - F(z) - F(a) E OM(I)

z-a

(a E I).

Typical examples of operator concave function is the logarithm and the entropy function

7J{z) == -z log z. In fact, Nakamura and Umegaki [22] proved the operator concavity of." and introduced the operator entropy

H(A) == -Alog A ~ 0 for positive contraction A in B(H) (see also Davis [7]). In the Kubo-Ando theory, the following functions are operator concave: I(z) =

Imz, r(z) = zml E

octo, 00) and Fm(z) == zm(1 -

z) E OC[O,I]. Moreover, Fm gives

an bridge between OC[O,I]+ and OM(O,oo)+ via operator means, see [10]:

THEOREM 4.1. A map m

1-+

Fm define. an affine order-i.omorphi.m from

the connection. to nonnegative operator concave junction. on [0,1]. One of the outstanding properties of operator concave functions is so-called Jensen's inequality. For a unital completely positive map. on an operator algebra and a positive operator A, Davis [6] showed

.(F(A»:::; F(.(A»

for an operator concave

function F. By Stinespring's theorem, a completely positive map is essentially a map X

1-+

C- XC. For a nonnegative function I, note that I E OM[O,oo) if and only if'

IE OC[O,oo) cf.

[16]. So Jensen's inequality by Hansen [15] is C·/(A)C:::; I(C- AC)

for

IICII:::; 1, A

~

O.

For nonnegative I E OM[O,oo), there exists a connection mj I(z)

Imz.

Then, the transformer inequality implies C·/(A)C = C-(lmA)C:::; C·CmC· AC:::; ImC- AC = I(C· AC). Hansen and Pedersen [16] gave equivalent conditions that Jensen's inequality holds:

167

Fujii

For a continuou. real function F on [O,a),

THEOREM(Hansen-Pedersen).

the lollowing. are equivalent: For 0 :::; A, B < a, (1) C* F(A)O:::; F(O· AO) lor (2) FE 00[0, a)

and

(3) PF(A)P:::; F(PAP)

11011:::; 1,

F(O) lor

~

0, every projection P,

(4) C* F(A)O + D· F(B)D:::; F(C* AO + D· BD) lor 0·0 + D* D :::; 1.

In the Kubo-Ando theory, for I(z) = Imz E OM(O,oo)+, the transpose is r(z) = zml = z/(l/z). Adopting this definition for

I

E

OM(O,oo), we have r(z)

-zlogz = ,,(z) for I(z) = logz. In general, the transpose of

J

E

=

OM(O,oo) is just a

function satisfying the above equivalent conditions (see [2,14,16]):

THEOREM 4.2.

J(z) E OM(O,oo) il and only il r(z) E 00[0,00) and

reO) ~ o. This theorem suggests that one can generalize the Kubo-Ando theory dealing with OM(O, 00)+, see [14].

5. RELATIVE OPERATOR ENTROPY Now we introduce the relative operator entropy S(AIB) for positive operators

A and B on a Hilbert space. If A and B is invertible, then it is defined as S(AIB) == Al/210g(A-l/2BA-l/2)Al/2 = B 1 / 2 ,,(B- 1 / 2 AB- 1 / 2)B 1 / 2 •

The above formula shows that S(AIB) can be defined as a bounded operator if B is invertible. Moreover, S(AIB

+ £) is monotone decreasing

as

£

! 0 by log z

So, even if B is not invertible, we can define S(AIB) by

(3)

S(AIB) == s-lim S(AIB + £) _.1.0

if the limit exists. Here one of the existence conditions is (see [14]):

E

OM(O,oo).

168

Fujii

The ,trong limit in (3) ezid,

THEOREM 5.1.

if and only if there ezi," c

with c ~ tB - (logt)A

(t

> 1).

Under the existence, the following properties like operator means hold: right monotonousness:

B ~ 0 ~ S(AIB,) ~ S(AIO),

right lower continuity:

B ..

transformer inequality:

T- S(AIB)T

!B

~

S(AIB.. ) ! S(AIB), ~

S(? ATIT- BT).

Conversely, if an operator function S'(AIB) satisfies the above axioms, then there exist

f

E

OM(O, 00) and FE 00[0,00) with F(O)

~

0 such that

for invertible A, B ~ 0, so that the class of such functions S' is a generalization of that of operator means or connections, see [141. In addition, the relative operator entropy has entropy-like properties, e.g.:

if A = tAl

subadditivity:

S(A + BIO + D)

joint concavity:

S(AIB)

+ (1 - t)A2

and B = tB1

~

for a normal positive linear map

+ from a

S(AIO) + S(OID),

tS(A1IBt} + (1 - t)S(A2IB2)

+ (1 - t)B2

informational monotonity:

~

for 0 ~ t ~ 1.

+(S(AIB»

~

S(+(A)I+(B»

W*-algebra containing A and B to a suitable

W*-algebra such that +(1) is invertible. In particular, we have Peierls-Bogoliubov inequality:

~(S(AIB» ~ ~(A)(log~(A) -log~(B»

for a normal positive linear functional ~ on a W*-algebra containing A and B. Now we apply S(AIB) to operator algebras. Let E be the conditional expectation of a type III factor M onto a subfactor /II, define a mazimal entropy S( /II ) as

S( /II ) == sup{IIS(AIE(A»1I1 A E Mt }. Then, for Jones' index [M : /11], we have (see [13))

Fujii

169 THEOREM 5.2.

Let.N be a .ubfactor of a type III factor M. Then, S( .N ) = 10g[M : .Nj.

Here we recall the relative entropy bra,

S(~I1/I)

~,

for states

1/1 on an operator alge-

ct., [27]. Derived from the Kullback:-Leibler information (divergence):

.. for probability vectors p, q,

'LP"log" PIt

"=1

q"

Umegaki [26] introduced the relative entropy S( ~11/I) for states ~, 1/1 on a semi-finite von Neumann algebra, which is defined as S(~I1/I)

= T(AlogA -

A log B)

where A and B are density operators of ~ and 1/1 respectively, i.e., ~(X)

= T(AX)

and

= T(BX).

1/I(X)

Araki [3] generalized it by making use of the Tomita-Takesaki theory, Uhlmann

[25] by the quadratic interpolation and Pusz-Woronowicz [24] by their functional calculus. These generalizations are all equivalent. The constructions of the last two entropies are based on the Pusz-Woronowicz calculus:

Put positive sesquilinear forms

+(X, Y) = ~(X*Y) and 'P(X, Y) = 1/I(XY*), then

According to these definition, we see some constructions of S(AIB). Making Ilse of the fact (z:' -l)/t! logz: as t! 0, we have (see [11,12])

Uhhnann type:

S(AIB) = s-lim Ag,B - A. t! 0

t

where g, is the operator mean satisfying 19,z: = z:'. Note that this formula gives an approximation of S(AIB). Putting +(z:, 1/)

=< Az:, 1/ > and 'P(z:,1/) =< Bz:, 1/ >, we have

170

Fujii

< 8(AIB)z,y >=

Pusz-Woronoiwicz type:

+

-(+log .)(z,y).

In [18), S.Izumino discussed quotient of operators by making use of Douglas' majorization theorem [8) and the parallel sum, which is considered as a space-free version of the PuszWoronowicz method. By making use of this, we also construct 8(AIB): Let R = (A B)1/2. Then, there exist X

+

and Y with X R = A 1/2 and Y R = B1/2, which are

uniquely determined by kernel conditions ker Reker X n ker Y. Here X· X Y·Y. Then, for F(z)

+ Y·Y is the projection onto ran Rand X· X commutes with

= 8(zll- z) = -zlog(zj(l- z», we have

Izumino type:

8(AIB) = R(F(X· X»R.

Recently, Hiai discussed a bridge between the relative entropy and the relative operator entropy. Note that if the density operators A and B commute, then

ffiai and Pet. [17) pointed out that the last term

had already been discussed by Belavkin and Staszewski [4). Hiai and Pet. showed that

for states on a finite dimensional C·-algebra. Furthermore, Hiai informed us by private communication that it also holds for states defined by trace class operators.

171

Fujii

REFERENCES [1) W.N .Anderson and R.J .Duffin: Series and parallel addition of matrices, J. Math. Anal. Appl., 28(1969), 576-594. [2) T.Ando: Topics on operator inequalities, Hokkaido Univ. Lecture Note, 1978. [3) H.Araki: Relative entropy of states of von Neumann algebras, Publ. RIMS, Kyoto Univ., 11 (1976), 809-833. [4) V.P.Belavkin and P.Staszewski: C*-algebraic generalillation of relative entropy and entropy, Ann. Inst. H. Poincare Sect. A.3'1(1982), 51-58. [5) J .Bendat and S.Sherman: Monotone and convex operator functions, Trans. Amer. Math. Soc., '19 (1955), 58-71. [6) C.Davis: A Schwarll inequality for convex operator functions, Proc. Amer. Math. Soc., 8(1957), 42-44. [7) C.Davis: Operator-valued entropy of a quantum mechanical measurement, Proc. Jap. Acad., 3'1(1961), 533-538. [8) R.G.Douglas: On majorillation, factorisation and range inclusion of operators in Hilbert space, Proc. Amer. Math. Soc., 1'1(1966), 413-416. [9) P.A.Fillmore and J.P.Williams: On operator ranges, Adv. in Math., '1(1971),254-281. [10) J .I.Fujii: Operator concave functions and means of positive linear functionals, Math. Japon., 25 (1980),453-461. [11) J.I.Fujii and E.Kamei: Relative operator entropy in noncommutative information theory, Math. Japon., 34 (1989), 341-348 . .(12) J.I.Fujii and E.Kamei: Uhlmann's interpolational method for operator means. Math. Japon., 34 (1989), 541-547. [13) J.I.Fujii and Y.Seo: Jones' index and the relative operator entropy, Math. Japon., 34(1989), 349-351. [14) J .I.Fujii, M.Fujii and Y.Seo: An extension of the Kubo-Ando theory: Solidarities, Math. Japon, 35(1990), 387-396. (15) F.Hansen: An operator inequality, Math. Ann., 248(1980), 249-250. [16) F.Hansen and G.K.Pedersen: Jensen's Inequality for operators and Lowner's theorem, Math. Ann., 258(1982), 229-241. [17) F.Hiai and D.Pets: The proper formula for relative entropy and its asymptotics in quantum probability, Preprint. [18) S.IlIumino: Quotients of bounded operators, Proc. Amer. Math. Soc., 108(1989), 427-435. [19) F.Kraus: Uber konvexe Matrixfunctionen, Math. Z., 41(1936), 18-42. [20) F.Kubo and T.Ando: Means of positive linear operators, Math. Ann., 248 (1980) 205-224. [21) K.Lowner: Uber monotone Matrixfunctionen, Math. Z., 38(1934), 177-216. [22) M.Nakamura and H.Umegaki: A note on the entropy for operator algebras, Proc. Jap. Acad., 3'1 (1961), 149-154. [23) W.PUSII and S.L.WoronowiclI: Functional calculus for sesquilinear forms and the purification map, Rep. on Math. Phys., 8 (1975), 159-170.

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Fujii

[24] W.Pusz and S.L.Woronowicz: Form convex functions and the WYDL and other inequalities, Let. in Math. Phys., 2(1978), 505-512. [25] A.Uhlmann: Relative entropy and the Wigner-Yanase-DY80n-Lieb concavity in an interpolation theory, Commun. Math. Phys., 54 (1977), 22-32. [26] H.Umegaki: Conditional expectation in an operator algebra IV, Kodai Math. Sem. Rep. 14 (1962), 59-85. [27] H.Umegaki and M.Ohya: Entropies in Quantum Theory (in Japanese), Kyoritsu, Tokyo (1984).

Department of Arts and Sciences (Information Science), Osaka Kyoiku University, Kasiwara Osaka 582 Japan MSC 1991: Primary 94A17, 47A63 Secondary 45B15, 47A60

173

Operator Theory: Advances and Applications, Vol. 59 @ 1992 Birkhauser Verlag Basel

AN APPLICATION OF FURUTA'S INEQUALITY TO ANDO'S THEOREM

Masatosm Fujii·, Takayuki Furuta •• and Eizaburo Kamei •••

Several authors have given mean theoretic considerations to Furuta's inequality which is an extension of Lowner-Heinz inequality. Ando discussed it on the geometric mean. In this note, Furuta's inequality is applied to a generalization of Ando's theorem. 1. INTRODUCTION.

Following after Furuta's inequality, several operator inequalities have been presented in [1,3,4,5,6,9,12]. We now think that they have suggested us a new progress of Furuta's inequality. In particular, Ando's result in [1] has been inspiring us this possibility, cf. [3,5,6]. Here we state Furuta's inequality [7] which is the starting point in our discussion.

FURUTA'S INEQUALITY. on a Hilbert .pace. If A ? B ? 0, then

Let A and B be po.itive operator. acting

(1) and

(2) for all P," ? 0 and q? 1 VIla. (1

+ 2,.)q ? P + 2,..

174

Fujii et at.

If we take p = 21' and q = 2 in (2), then we have AP

(3)

~

(AP/2 BP AP/2)1/2

for all p ~ O. From the viewpoint ofthis, Ando [1] showed that for a pair of selfadjoint operators A and B, A ~ B if and only if the exponential version of (3) holds, i.e.,

for all p ~ O. So we pay attention to the exponential order due to Hansen [10] defined by e A ~ eB , and introduce an order among positive invertible operators which is just opposite to the exponential one. That is, A > B means log A ~ log B. We call it the chaotic order because log A might be regarded as degree ofthe chaos of A. Thus Ando's result in [1] is rephrased as follows:

THEOREM A. Let A and B be po,itive invertible operator,. Then the following condition, are equivalent :

(a) A> B. (b) The following inequality hold, for all p

~

0i

(3) (c) The operator function p

~

G(p) = A-P 9 BP

i, monotone decrea,ing for

0, where 9 i, the geometric mean. In this note, we first propose the following operator inequalities like

Furuta's one, which are improvements of the results in the preceding note [6] :

THEOREM 1.

Let A and B be po,itive invertible operator,. If A

>

B,

then

(4) and

(5) for all p, l'

~

O.

This is a nice application of Furuta's inequality and implies the monotonity of an operator function discussed in [3] (see the next section), which is nothing but an extension of Theorem A by Ando. As a consequence, we also obtain Furuta's inequality in case of 21'q ~ p + 21' under the chaotic order.

Fujii et at.

175

2. OPERATOR FUNCTIONS. Means of operators established by Kubo and Ando [13] fit right in with our plan as in [4,5,6]. A binary operation m among positive operators is called a mean if m is upper-continuous and it satisfies the monotonity and the transformer inequality T"(A m B)T $ T* AT m T* BT for all T. We note that if T is invertible, then it is replaced by the equality T*(AmB)T = T*ATmT"BT. Now, by the principal result in [13], there is a unique mean m. corresponding to the operator monotone function z' for 0 $

$ 1j

B

1 m. z = z'

for z

~

O. Particularly the mean 9 = ml/2 is called the geometric one as in the case of

scalars. In the below, we denote m(1+.)/(p+.) by m(p,.) for all p ~ 1 and B ~ O. Here we can state our recent result in [3], which is a nice application of Furuta's inequality.

THEOREM B.

II A

~

B

~

0, then

iB a monotone increaBing function, that iI,

lorp~1 and1',B,t~O.

On the other hand, we have attempted mean theoretic approach to Furuta's inequality in [2,8,11,12]. It is expressed as

(7) and equivalently

(7') under the assumption A ~ B, p ~ 1 and l' ~ O. However the argument in [12], [9] and [3] might say that the key point of Furuta's inequality should be seen as

(8) under the same assumption.

176

Fujii et al.

Concluding this section, we state that (4) and (5) are rephrased to

(4') and

(5') respectively. H we take p = 2,. in (5'), then it is just (3) in Theorem A. 3. FURUTA'S TYPE INEQUALITIES. In this section, we prove Furuta's type inequalities (4) and (5) in Theorem 1.

We will use Ando's result (3). Moreover we need the following lemma on a mean m" cf.

[9]. LEMMA 2.

Let C and D be po.itif1e inf1ertible opemtor. and 0

:5 • :5 1.

Then

(a) C m, D = D m1-, C, = (D- 1 m, C- 1 )-1,

(b) D m, C and con.equentl1l

(c) Cm, D=(D- 1 m1-, C- 1)-1. PROOF. The function 1... (2) = 1 m 2 is called the representing function for a mean m and the map : m -+ I ... is an afline isomorphism of the means onto the operator monotone functions by [13]. Therefore it suffices to check that

for

2

> o.

Actually we have easily

by the transformer 'equality', and 1 m,

2

=

2

' = (-,)-1 2 = (1 m,

2

-1)-1 •

PROOF OF THEOREM 1. Assume that A:;» B. Then it follows from Theorem A that C = AP ~ (AP/ 2 BP AP/ 2)1 / 2 = D.

Suppose that 2,. ~ p ~ 0 and take t ~ 0 with 2,. ensures that

= p(1 + 2t).

Furuta's inequ&:lity (7')

Fujii et aJ.

177

In other words, and so

A27

~

(A7 BP A 7)27/(P+27). ~

Next we have to show the case where p

27'

>

O. Since B- 1

~

A-1,

Theorem A also implies that

that is,

(B7 A27 B7 )1/2

~

B27.

Again applying Furuta's inequality (7) to this, we have

and so

B- 27 (1+ 21 ) m (2,21) A27

> _I.

If we choose t ~ 0 with p = 27'(1 + 2t) since p ~ 27' ~ 0, then 1 - (1 27'/(p + 27') and consequently it is equivalent to

+ 2t)/(2 + 2t)

=

by Lemma 2 (c). This completes the proof. The following corollary of Theorem 1 plays an important role in the next section.

If A

COROLLARY 3.

B I then

(B7 AP B 7 )'/(P+27) ~ B'

(9) for p

~

~

0 and 27'

~ B ~

0 I and

(10) for all 7'

~

0 and p

~ B ~

O.

4. AN APPLICATION TO ANDO'S THEOREM. Finally we discuss a generalization of Theorem A by Ando [1]. Such an attempt has been done by [3], cr. also [9]. The purpose of this section is to complete it. A modification of Theorem B might be considered as in [3]. Let us define m(p,.,I)

for p

~

t

~

0 and

B ~

O. Clearly

= m(H.)/(p+.)

m(p,.,l)

= m(p,.)'

178

Fujii et al.

THEOREM 4.

ill monotone increalling for p

PRO 0 F . for p ~

8

~

~

B, then for a given t

t and r

~

~

0

o.

First of all, we prove thatfor a fixed r

> o. Putting m M t (p+8,r)

If A

> 0,

M t (P+8, r) ~ M t (p, r)

= m(p+,,2?,t) , it follows from (10) that

= B- 2? m

AP+'

= AP/2(A-p/2 B- 2? A-p/2 m A')AP/2 ~

AP/2«AP/2 B2? AP/2)-1 m (AP/2 B2? AP/2),/(p+2?»AP/2

= AP/2(A -p/2 B- 2?A -p/2)(p-t)/(p+2?) AP/2

p B-2? = A m(p-t)/(p+2?) -2? m(p,2?,t)· AP = B

The last equality is implied by Lemma 2 (a). Next we show the monotonity on r. Putting m = m(p,2?+"t) for 2r it follows from (9) that Mt(p,r

+ 8/2) =

~ 8 ~

0,

B-?(B-' m B1' AP B?)B-?

~

B-?«B? APB?)-,/(P+2?) m B? APB?)B-?

= B-? (B? AP B?)(H2?)/(p+2?) B-? = Mt(p,r).

As a result, Theorem A has the following generalization.

THEOREM 5.

For pOllitive invertible operatorl A and B, the following

condition, are equivalent : (a)A~B.

(b) For each jized t ~ 0, Mt(p, r) ~ At for r ~ 0 and p ~ t. (c) For each jized t ~ 0, Mt(p, r) ill a monotone increalling function for r ~ 0 and p ~ t.

Finally, we mention that Furuta's inequality is extended to the following in the sense of (8). Actually, if we take t = 1 in (b) of Theorem 5, then we have :

COROLLARY 8. (8)

If A ~ B, then (8) holdll, that ill,

179

Pujii et al.

REFERENCES

[1] T.Ando, On ,ome operator inequalitiu, Math.Ann., 279 (1987), 157-159. [2] M. Fujii, Furuta', inequality and it, mean theoretic approach, J. Operator Theory, 23 (1990), 67-72. [3] M.Fujii, T.Furuta and E.Kamei, Operator function, allociated with Furuta', inequality, Linear Alg. its Appl., 149 (1991), 91-96. [4] M.Fujii and E.Kamei, Furuta', inequality for the chaotic order, Math.Japon., 36 (1991), 603-606. [5] M.Fujii and E.Kamei, Furuta', inequality for the chaotic order, II, Math. Japon., 36 (1991),717-722. [6] M.Fujii and E.Kamei, Furuta', inequality and a generalization of Ando', theorem, Proc. Amer. Math. Soc., in press. [7] T.Furuta, A;::: B ;::: 0 allure, (B? AP B?)l/q ;::: B(P+2?)/q for l' ;::: O,p ;::: 0, q;::: 1 with [8]

(9] "0]

ill]

1.t2] 113]

(1 + 21')q;::: P + 2,., Proc.Amer.Math.Soc., 101 (1987), 85-88. T.Furuta, A proof via operator mean, of an order pre,erving inequality, Linear Alg. its Appl., 113 (1989), 129-130. T.Furuta, Two operator function, with monotone property, Proc.Amer.Math.Soc., 111 (1991), 511-516. F .Hansen, Selfadjoint mean, and operator monotone function" Math.Ann., 256 (1981), 29-35. E. Kamei, Furuta', inequality via operator mean, Math.Japon., 33 (1988), 737-739. E. Kamei, A ,atellite to Furuta', inequality, Math.Japon., 33 (1988), 883-886. F.Kubo and T.Ando, Mean, of po,itive linear operator" Math.Ann., 246 (1980), 205-224.

• Department of Mathematics, Osaka Kyoiku University, Tennoji, Osaka 543, Japan •• Department of Applied Mathematics, Faculty of Science, Science University of Tokyo, Kagurazaka, Shinjuku, Tokyo 162, Japan .... Momodani Senior Highschool, Ikuno, Osaka 544, Japan

MSC 1991: Primary 47A63

Secondary 47B15

180

Operator Theory: Advances and Applications, Vol. 59 © 1992 Birkhiiuser Verlag Basel

APPLICATIONS OF ORDER PRESERVING OPERATOR INEQUALITIES TAKAYUKI FURUTA

°

A ;::: B ;::: assures (Br AP Br)l/q ;::: B(p+2r)/q for r ;::: 0, p ;::: 0, q ;::: 1 with (1 + 2r)q ;::: (p + 2r). This is Furuta's ineql\~lity. In this paper, we show that Furuta's inequality can be applied to estimate the value of the relative operator entropy and also this inequality can be applied to extend Ando's result.

§o.

INTRODUCTION

An operator means a bounded linear operator on a complex Hilbert space. In this paper, a capital letter means an operator. An operator T is said to be positive if

(Tx, x) ;:::

°for all x in a Hilbert space. We recall the following famous inequality

A ;::: B ;::: 0, then All! ;::: BIl! for each

0

j

if

E [0,1]. This inequality is called the Lowner-

Heinz theorem discovered in [14] and [12]. Moreover nice operator algebraic proof was shown in [16]. Closely related to this inequality, it is well known that A ;::: B ;:::

°does

not always ensure AP ;::: BP for p > 1 in general. As an extension of this Lowner-Heinz theorem, we established Furuta's inequality in [7] as follows; if A ;::: B ;::: 0, then for each r;::: 0,

and

hold for each p and q such that p ;::: 0, q ;::: 1 and (1

+ 2r)q

;::: p

+ 2r.

We remark

that Furuta's inequality yields the Lowner-Heinz theorem when we put r = 0. Also we remark that although AP ;::: BP for any p > 1 does not always hold even if A ;::: B ;::: 0, Furuta's inequality asserts that J(AP) ;::: J(BP) and g(AP) ;::: g(BP) hold under the suitable conditions where J(X)

= (WXW)l/q

and g(Y)

= (ArYAr)l/q.

Alternative

proofs of Furuta's inequality are given in [4][8][9] and [13]. The relative operator entropy for positive invertible operators A and B is defined in [2] by

SeA I B)

= Al/2(logA-l/2BA- 1/ 2)Al/2.

Furuta

181

In [11], we showed that Furuta's inequality could be applied to estimate the value of this relative operator entropy S(A invertible operators. Then logG

~

I B).

logA

~

For example, let A,B and G be positive

logB holds if and only if

holds for all p ~ 0 and all r ~ O. In particular logG ~ logA- 1 ~ logB ensures S(A I G)

~

-2AlogA

~

S(A I B) for positive invertible operators A, Band G. In this

paper, we shall attempt to extend this result by using Furuta's inequality. In [11], we showed an elementary proof of the following result which is an extension of Ando's one [1]. Let A and B be selfadjoint operators. Then A if for a fixed t

~

~

B holds if and only

0,

is an increasing function of both p and r for p

~

t and r

~

O. In this paper, also by

using Furuta's inequality we shall attempt to extend this result. §1.

APPLICATION TO THE RELATIVE OPERATOR ENTROPY

We shall show that Furuta's inequality can be applied to estimate the value of the relative operator entropy in this section. Recently in [2], the relative operator entropy S(A I B) is defined by

for positive invertible operators A and B. We remark that S(A usual operator entropy. This relative operator entropy S(A

I B)

I I) =

-AlogA is the

can be considered as

an extension of the entropy considered by Nakamura and Umegaki [15] and the relative entropy by Umegaki [17].

THEOREM 1. Let A and B be positive invertible operators. Then the following assertions are mutually equivalent. (I) logA

~

10gB.

(110) AP ~ (AP/2BPAP/2)1/2 for allp ~ O.

(III) AP ~ (AP/2 B S AP/2)p/(p+s) for all p ~ 0 and all s ~

o.

182

Furuta

(II2) AP 2: (AP/2 BSO AP/2)p/(p+so) for a fixed positive number So and for all p such that p E [0, Po], where Po is a fixed positive number. (113) APo 2: (APo/2 BS APo/2)Po/(Po+s) for a fixed positive number Po and for all s such that s E [0, so], where So is a fixed positive number. (1111) logAP+s 2: log(AP/2 B SAP/2) for all p 2: 0 and all s 2: O. (1112) logAP+sO 2: log (AP/2 B 8 0 AP/2) for a fixed positive number So and for all p such that p E [O,Po], where Po is a fixed positive number.

THEOREM 2. Let A and B be positive invertible operators. Then the following assertions are mutually equivalent.

(I) logC 2: logA 2: logB (110) (AP/2Cp AP/2) 1/2 2: AP 2: (AP/2 BP AP/2) 1/2 for all p 2:

o.

(lId (AP/2C8 AP/2)p/(p+s) 2: AP 2: (AP/2 BS AP/2)p/(p+s) for all p 2: 0 and all s 2: O. (112) (AP/2c soAP/2)p/(p+ so) 2: AP 2: (AP/2 BSo AP/2)p/(p+so) for a fixed positive number So and for all p such that p E [O,Po], where Po is a fixed positive number. (113) (APo/2cs APo/2)Po/(Po+s) 2: APo 2: (APo/2 BS APo/2)Po/(Po+s) for a fixed positive number Po and for all s such that

8

E [0, so], where 80 is a fixed positive number.

(IIII) log(AP/2cs AP/2) 2: logAP+s 2: log(AP/2 BS AP/2) for all p ~ 0 and all s 2: O. (1112) log(AP/2c soAP/2) ~ logAP+sO 2: log(AP/2 BSo AP/2) for a fixed positive number 80 and for all p such that p E

(IVI) S(A-P I CS) 2: S(A-P I AB) (IV2) S(A-P

I cso) 2:

S(A-P

[0, Po], where Po is a fixed positive number. ~

I ASo)

S(A-P I B S) for all p ~

S(A-P

I B80 )

~

0 and all s

~

o.

for a fixed positive number

80

and

for all p such that p E [0, Po], where Po is a fixed positive number.

COROLLARY 1 [11]. Let A, Band C be positive invertible operators. If logC 2: logA-I 2: logB, then S(A I C) ~ -2AlogA 2: S(A I B).

In order to give proofs to Theorem 1 and Theorem 2, we need the following Furuta's inequaliy in [7].

183

fUruta

THEOREM A (Furuta's inequality). Let A and B be positive operators on a

B

~

0, then

1 and r

~

0.

Hilbert space. If A

~

(i) and

(ii) hold for all p

~

For any real

Let A and B be invertible positive operators.

LEMMA 1 [10]. number r,

LEMMA 2. Let A and B be positive invertible operators. Then for any p, 8

~

0,

the following assertions are mutually equivalent. ~

(i)

AP

(AP/2 B8 AP/2)P/(8+P) ,

(ii)

(B8/2 AP B8/2)8/(8+p)

~

BS.

Proof of Lemma 2. Assume (i). Then by Lemma 1, AP

~

(AP/2 B SAP/2)p/(8+p) = AP/2 Bs/2(Bs/2 AP B8/2)-8/(8+P) B8/2 AP/2,

,that is,

holds. Taking inverses proves (ii). Conversely, we have (i) from (ii) by the same way. Proof of Theorem 1. (I)

(110) is shown in [1]. (lId

~

(110) is obvious by putting s=p in (lId. We show (110) ~ (lId. Assume (110); AS ~ (As/2 B SAS/ 2)1/2 for all (I)

8 ~

0,

and Po ~ 0,

F(p, ro) = B-ro(Bro AP B ro)(t+2ro)/(p+2ro)B-ro is an increasing function of p such that p E [0, Po] for P ~ t. (112) For any fixed t ~ 0, ro

> 0,

and Po ~ t,

F(PQ, r) = B-r(Br APo Br)(t+2r)/ 0,

and Po ~ 0,

G(p, ro) = A-ro(AroBP Aro)(t+2ro)/(p+2ro)A-ro is a decreasing function of p such that p E [0, Po] for P ~ t. (1112) For any fixed t ~ 0, ro

> 0, and Po

~ t,

G(PQ, r) = A-r(Ar BPo A r )(t+2r)/(Po+2r) A-r is a decreasing function of r such that

r E [O,ro]. (IV) For any fixed t

~

0, and r

~

0,

log(Br AP B r )(t+2r)/(P+2r) is an increasing function of p for p ~ t.

187

Furuta

(V) For any fixed t

~

0, and r

~

0,

log(Ar BP Ar)(t+2r)/(p+2r) is a decreasing function of p for p ~ t. Proof of Theorem 3. (I)

===?

(110). Assume (I). First of all, we cite (10) by (I)

and (lId of Theorem 1

AP ~ (AP/2 B2r AP/2)p/(p+2r) for all p ~

(10)

°and all r

~

O.

Moreover (10) ensures the following (ll) by the Lowner-Heinz theorem

AS

(11)

~

(AP/2 B2r AP/2)s/(p+2r) for all p ~ s ~ 0 and all r ~ O.

Then by (11), we have

(Br AP B r )(p+s+2r)/(p+2r) = B r AP/2(AP/2 B2r AP/2)s/(p+2r) AP/2 B r

by Lemma 1

by (ll)

So the following (12) and (13) hold for each r

~

°

and each p

~

s

~

0 ,

(12)

and (13) (13) is an immediate consequence of (12) because logB- 1 ~ logA- 1 ensures that

(A -r B-1! A-r) (p+s+2r)/(p+2r) :5 A -r B-(P+S) A-r holds for each r

~

0 and for each p and s such that p

~

s

~

0 . Taking inverses gives

(13). As (t+2r)/(p+s+2r) E [0,1] since p ~ t ~ 0, (12) ensures the following inequality

by the Lowner-Heinz theorem

(Br AP+s Br)(t+2r)/(p+s+2r)

~

which implies the following results for a fixed t

(14)

(Br AP Br)(t+2r)/(p+2r) , ~ 0

and r

~ 0

(Br AP Br)(t+2r)/(p+2r) is an increasing function of p ~ t,

&Ild

(15)

(Ar BP Ar)(t+2r)/(p+2r) is a decreasing function of p ~ t,

because (15) is easily obtained by (13) &Ild its proof is the same way as in the proof of (14) from (12).

188

Furuta

Next we show the following inquality (16) (16) (AP/2 B 2r AP/2)(t-p)/(2r+p)

~

(AP/2 B2s AP/2)(t-p)/(2s+p)

for r

~ 8 ~

O.

By (15), we have

(17) (AP/2 B2r AP/2)(tl+p)/(2r+p) :5 (AP/2 B2s AP/2)(t 1 +P)/(2s+p) Put 0 = (p - t)/(p theorem, taking

0

+ tl)

E [0,1] since p ~ t ~

°and tl

~

for 2r

~

28 ~ tl ~ 0.

O. By the Lowner-Heinz

as exponents of both sides of (17) and moreover taking inverses of

these both sides, we have (16). Therefore for r

~ 8 ~

0,

F(p, r) = B-r(Br AP B r )(t+2r)/(p+2r) B-r = AP/2(AP/2 B2r AP/2)(t-p)/(p+2r) AP/2 ~

by Lemma 1

AP/2(AP/2 B2s AP/2) (t-p)/(p+2s) AP/2

by (16)

=B-S(B SAP B S)(t+2s)/(P+2s) B-s

by Lemma 1

=F(p, 8), so we have (110) since F(p, r) is an increasing function of p (I)

==}

~

t by (14). So the proof of

(110) is complete.

(110)

(lIt) and (110)

==}

==}

(112) are obvious since both (lIt) and (112) are special

cases of (110)' (lIt)

(I) . Assume (III)' Then F(p, ro) ~ F(O, ro) with t = 0, that is,

==}

equivalently, by Lemma 1 namely (18) holds for all p such that p E [0, Po] and a fixed ro > 0. Taking logarithm of both sides of (18) since logt is an operator monotone function, we have

(p + 2ro)logA ~ log(AP/2 B2ro AP/2). Letting p

(112)

--+

==}

0, we have logA ~ logB since ro is a fixed positive number.

(I). Assume (112)' Then FCPo, r) ~ FCPo,O) with t

= 0, that is,

Furuta

189

equivalently,

(19) for all r such that r E [0, ro] and a fixed PO

> O. Taking logarithm of both sides of (19)

since logt is an operator monotone function, we have

Letting r

--+

0, we have logA

~

logB since PO is a fixed positive number.

(I) ==> (IIIo). This is in the same way as (I) ==> (110). (IIIo) ==> (lIlt) and (IIIo) ==> (I1I2) are obvious since both (lIlt) and (1112) are special cases of (IIIo). (lIlt) ==> (I) and (I1I2) ==> (I) are obtained by the same ways as (lIt) ==> (I) and (112) ==> (I) respectively. (110) ==> (IV) and (IIIo) ==> (V) are both trivial since logt is an operator monotone function.

(IV) ==> (I). Assume (IV) with t = O. Then

that is,

Letting r

--+

0 and p = 1, we have logA

~

logB.

(V) ==> (I). This is in the same way as (IV) ==> (I). Hence the proof of Theorem 3 is complete. We remark that the equivalence relation between (I) and (110) has been shown in [6] as an extension of [5,Theorem 1]. I would like to express my sincere appreciation to Professor T. Ando for inviting me to WOTCA at Sapporo and his hospitality to me during this Conference which has been held and has been excellently organized during June 11-14,1991. I would like to express my cordial thanks to the referee for reading carefully the first version and for giving to me useful and nice comments.

190

Furuta

References [1] T.Ando, On some operator inequality, Math. Ann.,279(1987),157-159. [2] J.I.Fujii and E.Kamei, Relative operator entropy in noncommutative information theory, Math. Japon.,34(1989),341-348. [3] J.I. Fijii and E.Kamei, Uhlmann's interpolational method for operator means, Math. Japon.,34(1989),541-547. [4] M.Fujii, Furuta's inequality and its mean theoretic approach, J. of Operator Theory,23(1990),67-72. [5] M.Fujii, T.Furuta and E.Kamei, Operator functions associated with Furuta's inequality, Linear Alg. and Its Appl., 149(1991),91-96. [6] M.Fujii, T.Furuta and E.Kamei, An application of Furuta's inequality to Ando's theorem, preprint. [7] T.Furuta: A ~ B ~ 0 assures (Br APBr)l/q ~ B(p+2r)/q for r ~ O,p ~ O,q ~ 1 with (1 + 2r)q ~ (p + 2r). Proc. Amer. Math. Soc., 101(1987),85-88. [8] T.Furuta, A proof via operator means of an order preserving inequality, Linear Alg. and Its Appl.,113(1989),129-130. [9] T.Furuta, Elementary proof of an order preserving inequality, Proc. Japan Acad.,65(1989),126. [10] T.Furuta, Two operator functions with monotone property, Proc. Amer. Math. Soc.,111(1991),511-516. [11] T.Furuta, Furuta's inequality and its application to the relative operator entropy, to appear in J. of Operator Theory. [12] E.Heinz, Beitragze zur Sti'ungstheorie der Spektralzerlegung, Math. Ann.,123(1951),415-438. [13] E.Kamei, A satellite to Furuta's inequality, Math. Japon,33(1988),883-886. [14] K.L6wner, Uber monotone Matrixfunktion, Math. Z.,38(1934),177-216. [15] M.Nakamura and H.Umegaki, A note on the entropy for operator algebras, Proc. Japan Acad.,37(1961),149-154. [16] G.K.Pedersen, Some operator monotone functios, Proc. Amer. Math. Soc.,36(1972),309-31O. [17] H.Umegaki, Conditional expectation in operator algebra IV, (entropy and information), Kadai Math. Sem. Rep.,14(1962),59-85

Department of Applied Mathematics Faculty of Science Science University of Tokyo 1-3 Kagurazaka, Shinjuku Tokyo 162 Japan MSC 1991: Primary 47A63

191

Operator Theory: Advances and Applications, Vol. 59 © 1992 Birkhiiuser Verlag Basel

THE BAND EXTENSION ON THE REAL LINE AS A LIMIT OF DISCRETE BAND EXTENSIONS, I. THE MAIN LIMIT THEOREM I. Gohberg and M.A. Kaashoek

In this paper it is proved that the band extension on the real line (viewed as convolution operator) may be obtained as a limit in the operator norm of block Laurent operators of which the symbols are band extensions of appropriate discrete approximations . of the given data. 1.

O. INTRODUCTION

Let k be an m x m matrix function with entries in L2([-T,T]). An m x m matrix function

f with entries in Ll(R) n L2(R)

(a) f(t) = k(t) for

-T

is called a positive extension of kif

~ t ~ T,

(b) I -1(>.) is a positive definite matrix for each A E R. Here

1 denotes the Fourier transform of f.

If (b) is fulfilled, then

(I -f

II=-<X>

00

(4.3)

v=-oo

-00

The proof of Proposition 4.1 is the same as that of Lemma 1.5.1 in (9) and, therefore, it is omitted. Here we only note that for v =

0, ±1, ±2, ... the operator F" in

Proposition 4.1 is the Hilbert-Schmidt operator on L2'([O, O'D defined by

(4.4)

(F"cp)(t) =

l

u

f(t-s+vO')cp(s)ds,

O:St:SO'.

1.5 The algebra B( r). Throughout this section r >

°is a fixed posi-

tive number. By B(r) we denote the set of bounded linear operators T on L2'(R) such that the r-block partitioning of T is a L2'([O, rD-block Laurent operator with symbol in W(S2,L2'([O,r);T). Take T E B(r), and let (Ti-j)i,j=_oo be its r-block partitioning. We define:

(5.1a)

E

IIITIII:=

IITjl1
00

(by (3.1) and (2.20».

216

Gohberg and Kaashoek

Next, note that

Here we used that for each T E B( T)

because of (2.4) and the inequality (2.8) for Qd(n). By (2.9a) and (2.4) we have

IIQd(n)L",IIB(T) ...... 0 if n ......

00.

Thus (3.6) yields (n ...... (0),

and therefore (3.8) From (3.6), (3.8) and the identities (3.2) and (3.4) we conclude that

111.4 Discrete and continuous orthogonal functions.

Let k be an

m x m matrix function with entries in L2([ -T, T]), and let K be the operator on L2'([O, T]) defined by

(Kcp)(t) =

loT k(t -

s)cp(s)ds,

0:5 t :5

T.

In what follows we assume that 1 - K is invertible. The corresponding resolvent kernel is denoted by -yet,s), i.e.,

(4.1)

.«1 - K)-lcp)(t) = cp(t) + loT -y(t,s)cp(s)ds,

0:5 t :5

T.

By definition (see [14], [15]) the (first kind) orthogonal function associated with k is the entire m x m matrix function D(·) defined by

(4.2)

~E

c.

217

Gohberg and Kaashoek

Next, let n be an arbitrary positive integer, and let

(I -Kln) -Kin)

(4.3)

-K~~) I -

_K(n) n-l

be the ~-block partitioning of I -

K'n)

-(n-l) _K(n) -(n-2)

K~n)

_K(n) n-2

1-

Ka n )

In particular, K~n) (v

K.

)

= -(n -

1), ... , n - 1) is the

operator on L2'([O, ~r]) defined by

(K~n)'P)(t)

=

1* o

k(t -

8

v + -r)'P(8) d8, n

0< - t -< ~. n

Since I - K is invertible, the n X n operator matrix in (4.3) is invertible. By definition (see [1], [13]), the orthogonal polynomial associated with (4.3) is the operator polynomial

Pn(z) given by (4.4) where

r L~J ~ -i~~, 1+

0 K'n) -Kin)

Xo

Xl

(4.5)

We shall see that for n

-1

I - K~n) _K(n) n-2

--+ 00

_K(n) -(n-2)

C)

I _ ~~n)

0

_K~(~_'))

_K(n)

-

0

..

the n-th orthogonal polynomial P n converges

in a certain sense to the orthogonal function D associated with k. To make this statement precise, let Tn be the operator on L2'(R) whose ~-block partitioning is the L2'([O, ~r])­ block Laurent operator with symbol Lj;~ zjXj . Since k has its entries in L 2 ([-r,r]), the operators K~n) (v

(4.5)

(4.6)

= -(n -

1), ... , n - 1) are Hilbert-Schmidt operators. We may rewrite

as

UJ

(I -Kln) =

-1

_K~n)

1 - Ka n )

_K(n)

_K(n)

n-l

-(n-l) )( Ko,n») K,n) K(n)

_K(n)

n-2

-

_K(n) -(n-2) I

-

~(n) 0

1

K(:n) n-l

.

Gohberg and Kaashoek

218

It follows (by the same arguments as used in the proof of Proposition 11.1.1) that Xo, ... ,Xn -

are Hilbert-Schmidt operators, and thus (cf., Section 1.5) the operators Tn

1

belong to the algebra B(~T). Let Zn

be the kernel function of Tn. From (4.6) we see that

Zn

is given by

(4.7a)

0:5 t :5 T,

zn(t,s)=k(t-s)+ 1T' 'Y(t,u)k(u-s)du,

(4.7b)

zn(t,s) = 0,

(4.7c)

zn(t + -T,S + -T) n n

1

0:5 s :5

t E R\[O,T), 1

= zn(t,s),

0:5 s :5

1 n

-T,

1 n

-T,

(t,s)ER.

Now, put Z(t) = { 'Y(t, 0), 0,

0:5 t :5 T, t E R\[O, T).

Then we see from (4.1) that Z(t) = k(t)

(4.8)

+ 1T' 'Y(t,u)k(u)du,

o :5 t :5 T,

and hence the formulas (4.7a) - (4.7c) suggest that for n

-+ 00

the function zn(t,s) con-

verges to z(t - s). The precise result is the following. PROPOSITION 4.1. Let T, TJ, T2, ... be the integral operators on L2"(R)

with kernel [unctions z(t - s), Zl (t, s), Z2(t, s), . .. , respectively. Then

liT -

(4.9)

T(n)lIs(T')

PROOF. Note that T

= L.,.

-+

(n

0

Since

Z

-+

(0).

has its entries in LI(R) n LI(R), we

know (see Proposition 1.4.1) that T E B(T). Proposition 1.5.2 implies that Tn E B(T) for n

~

1. Next we use notations and results from the previous section. From the definition

of Z it follows that (4.10)

Furthermore, because of (3.6), we have (4.11)

Gohberg and Kaashoek

219

But then, by the arguments used in the proof ofthe limit formula (1.5) (see Section III.3), we obtain (4.9) . •

REFERENCES

[1]

[2] [3] [4] [5]

[6] [7] [8]

[9] [10)

[11] [12]

[13]

[14]

[15]

A. Atzmon, N-th orthogonal operator polynomials, in: Orthogonal matrixvalued polynomials and applications, OT 34, Birkhauser Verlag, Basel, 1988; pp. 47-63. D.Z. Arovand M.G. Krein, Problems of search of the minimum of entropy in indeterminate extension problems, Fun ct . .4.nal. Appl. 15 (1981), 123-~26. J.J. Benedetto, A quantative maximum entropy theorem fro the real line, Integral Equations and Operator Theory 10 (1987), 761-779. S. Bochner and R.S. Phillips, Absolutely convergent Fourier expansions for non commutative rings, Annals of Mathematics 43 (1942) 409-418. J. Chover, On normalized entropy and the extensions of a positive definite function. J. Math. Mech. 10 (1961),927-945. H. Dym, J contractive matrix functions, reproducing kernel Hilbert spaces and interpolation, CBMS 71, Amer. Math. Soc., Providence RI, 1989. H. Dym and I Gohberg, On an extension problem, generalized Fourier analysis and an entropy formula, Integral Equations Operator Theory, 3 (1980), 143-215. I. Gohberg, S. Goldberg and M.A. Kaashoek, Classes of Linear Operators, Volume I, Birkhauser Verlag, Basel, 1990. I. Gohberg and M.A. Kaashoek, Asymptotic formulas of Szego-Kac-Achiezer type, Asymptotic Analysis, 5 (1992), 187-220. I. Gohberg, M.A. Kaashoek and H.J. Woerdeman, The band method for positive and contractive extension problems, J. Operator Theory, 22 (1989), 109-105. I. Gohberg, M.A. Kaashoek and H.J. Woerdeman, The band method for positive and contractive extension problems: An alternative version and new applications, Integral Equations Operator Theory 12 (1989), 343-382. I. Gohberg, M.A. Kaashoek and H.J. Woerdeman, A maximum entropy pricipIe in the general framework of the band method, J. Funct, Anal. Anal. 95 (1991), 231-254. I. Gohberg and L. Lerer, Matrix generalizations of M.G. Krein theorems oon orthogonal polynomials, in: Orthogonal matrix-valued polynomials and applications, OT 34, Birkhauser Verlag, Basel, 1988; pp. 137-202. M.G. Krein, Continuous analogues of propositions about orthogonal polynomials on the unit circle (Russian), Dokl. Akad. Nauk USSR, 105:4 (1955), 637-640. M.G. Krein and H. Langer, On some continuation problems which are closely related to the theory of operators in spaces II",. IV, J. Operator Theory 13 (1985), 299-417.

220

Gohberg and Kaashoek

[16]

D. Mustafa and K. Glover, Minimum entropy Hoc control, Lecture Notes in Control and Information Sciences 146, Springer Verlag, Berlin, 1990.

I. Gohberg Raymond and Beverly Sackler Faculty of Exact Sciences School of Mathematical Sciences Tel-Aviv University Ramat-Aviv, Israel

M.A. Kaashoek Faculteit Wiskunde en Informatica Vrije Universiteit De Boelelaan 1081a 1081 HV Amsterdam The Netherlands

MSC: Primary 47A57, Secondary 45AlO, 47A20

221

Operator Theory: Advances and Applications, Vol. 59 @ 1992 Birkhiiuser Verlag Basel

INTERPOLATING SEQUENCES IN THE MAXIMAL IDEAL SPACE OF Hoo II Keiji Izuchi The objective of this paper is to study interpolating sequences in M(Hoo) the maximal ideal space of Hoo. Using the pseudohyperbolic distance, L. Carleson gave a characterization of interpolating sequences in D the open unit disk. But its condition does not characterize interpolating sequences in M(Hoo). Carleson's condition has several equivalent conditions in D. It is studied the relations between these conditions and interpolating sequences in M(Hoo). 1. INTRODUCTION

Let HOO be the Banach algebra of bounded analytic functions on the open unit disc D with the supremum norm. We denote by M(Hoo) the maximal ideal space of HOO with the weak-*topology. We identify a function in Hoo with its Gelfand transform. We may consider that DC M(Hoo), and by the corona theorem D is dense in M(Hoo). The maximal ideal space M(Loo) of Loo(8D) may be considered as a subset of M(Hoo). For points x and 11 in M(Hoo), we define

p(x,y)

= sup {lh(x)1 j h(y) = 0, hE Hoo,

IIhll ~ I}.

When z and w are points in D, p(z, w) = Iz - wi / 11 - zwl. For a point x in M(Hoo), we put P(x) = {y E M(Hoo)j p(y, x) < I}, which is called a Gleason part. If P{x) = {x}, x is called a trivial point. Every point in M(Loo) is trivial. If P(x) =F {x}, then x and P(x) are called nontrivial. In this case, there is a one to one continuous map L., from D onto P(x) such that x = L.,(O) and HOO 0 L., C HOO. Moreover if L., is a homeomorphism, P(x) is called a homeomorphic part. P(O) = D is a typical homeomorphic part. We put . G = {x E M(Hoo)j x is not nontrivial}. Then Gis an open subset of M(Hoo) (see [6]). For a sequence {z"},, in D with E:'=11-lz,,1 < 00, a function b(z) = IT _-_z_" """z_---...:z,,::... zED

,,=1 Iz,,1 1 - z"z'

222

lzuchi

is called a Blaschke product with zeros {zn}n. For a positive singular measure p on aD, a function

S[}.t](z) = exp(-

J

+

ei9 Z --9e' - Z

dp(B))

is called a singular inner function. For a bounded measurable function F on aD such that log IFI is integrable, a function

is called an outer function. f in Hoo is represented by

These functions are contained in Hoo, and every function OUog Ifl] S[p] b except a constant factor (see [5]). Let

f =

fn = OUog IfnI] S[Pn] bn be a function in HOO with IIfnll :::; 1. If E:=110g ifni is integrable, E:=1 Pn(aD) < 00 and II:=1 bn is still a Blaschke product, then we denote by II:=1 fn the function O[E:=110g Ifni] S[E:=1Pn] II:=1 bn in Hoo. A sequence {xn}n in M(HOO) is called interpolating if for every sequence {an}n of bounded complex numbers there is a function f in HOO such that f(xn) = an for all n. An interpolating sequence {zn}n in D is characterized by Carleson [2] as follows; inf k

II

n:n~k

I1Zk- -ZnZk Zn I > o.

Let

./, ( ) 'f/n Z

be a Blaschke factor.

-zn Z - Zn = ---_IZnl 1 - ZnZ Then 7/Jn(Zn) = 0, II7/Jnll = 1 and inf k

II

n:n~k

P(Zn,Zk)

=

infl( II 7/Jn)(Zk)l. k n:n~k

From this observation, we can consider similar conditions for sequences {xn}n in M(HOO) as follows: inf II p(xn' Xk) > k n:n~k

o.

There is a sequence {gn}n in HOO such that

(A3)

There is a sequence {fn}n in HOO such that

In this paper, we shall study what kind of relations are there between interpola.ting sequences and conditions (Ai), i = 1,2,3. First, we ha.ve

Izuchi

223

By the open mapping theorem,

FACT 2. If {x"},, is interpolating, {x"},, satisfies (A2). In Theorem 1, we shall prove that the converse assertion of Fact 2 is true when each point x" is trivial. A sequence {x"},, in M(HOO) is called strongly discrete if there is a sequence of disjoint open subsets {U"},, of M(HOO) such that x" E U". Then we have

FACT 3. If {x"},, is interpolating, {x"},, is strongly discrete. A Blaschke product is called interpolating if its zero sequence is interpolating. For a function f in Hoo, we put

Z(f)

= {x E M(HOO)j

f(x)

= O}.

Then we have

FACT 4 (see [8]). Let {x"},, be a strongly discrete sequence. If {x"},, C Z(b) for some interpolating Blaschke product b, then {x"},, is interpolating. In [4], Gorkin, Lingenberg and Mortini proved that if {x"},, is a sequence in a homeomorphic part and satisfies condition (AI), then there is an interpolating Blaschke product b such that {x"},, C Z(b). In [7], the author got the same conclusion when {x"},, is a sequence in a homeomorphic part and satisfies condition (A2). Also in [8], the author studied a sequence whose elements are contained in distinct parts. In Theorem 2, we shall prove that if {x"},, is a sequence in G, then {x"},, satisfies condition (A3) if and only if {x"},, is strongly discrete and there is an interpolating Blaschke product b such that {x"},, C Z(b). And in Theorem 3, we show that there is a strongly discrete sequence {x"},, in a nontrivial part such that {x"},, satisfies condition (AI) but {x"},, is not interpolating.

2. CONDITION (A 2 ) In this section, the following lemma plays an important role. LEMMA 1 [9, Theorem 3.1]. Let b be a Blaschke product. Then there are Blaschke products bl and b2 such that b = bl b2 and bl(x) = b2 (x) = 0 for every point x in

M(HOO) with bIP(s)

= o.

First we prove

PROPOSITION 1.

(A3) => (A 2 ) => (AI).

PROOF. The implication (A3) => (A:I) is trivial. To prove (A:I) => (Ad, it is

Izuchi

224

sufficient to prove that if I E Hoo satisfies

IT

n=1

11/11 :::; 1 and I(xn)

p(x, xn) ~ I/(x)1

= 0 for every n, then

for x E M(HOO).

Let I = OOog I/Il S[J.t] b be a canonical factorization. We devide {xn}n into three parts:

{(j}j

=

{xn; OOog I/Il S[Jl]IP(xn) = O};

{€j}j

= =

{xn; Xn

pj}j

f/.

{(j}j and bIP(xn)

= O};

{xn; I does not vanish indentically on P(xn)}.

Let k be an arbitrary positive integer. We put

(1)

By [6, Theorem 5.3], there are interpolating Blaschke products b1, b2 , • •• ,b k such that b = (117=1 bj ) Band bj(,\j) = 0 for j = 1,2, ... , k for some Blaschke product B. Then we have

(2) We apply Lemma 1 for B succeedingly. Then we have a factorization B Bj(€i) = 0 for every i. Hence

= IIj=IBj such that

By (2), we have

Therefore by (1),

Letting k

-+ 00,

we have II::"=1 p(x, xn) ~

I/(x)l.

Now we prove the following theorem. The idea of this proof comes from Axler and Gorkin [1, Theorem 3].

THEOREM 1. Let {xn}n be a sequence 01 trivial points in M(HOO). Then {xn}n is interpolating if and only if {xn}n satisfies condition (A 2 ).

Izuchi

225

PROOF. Suppose that {xn}n satisfies condition (A2). Then there is a sequence ~ 1,

{Fn}n in HOO such that IlFnll

=

(1)

Fn(XIc)

(2)

Fn(xn) '" 0

We may assume that Fn(xn) such that

> 0 for every n. Let {En}n be a sequence of positive numbers

0

if k '" n, and for every n.

(3) Take 0

< On < 1 such that

Here we note that if Fn(xn) closes to 1 then On closes to o. Let Fn = OnSnBn be a canonical factorization. For a positive integer k, we apply Lemma 1 for Bn k-times succeedingly. As a result, we have a factorization Bn = BnIBn,··· Bnt . Since Xj is a trivial point by our starting hypothsis, if Bn (x j) = 0 for some j then Bnl (x j) = ... = Bnt (x j) = o. Here we may assume that

Then by (2),

I(O!'lcS~'Ic Bnt)(Xn)1 ~ Fn(xn)1fIc > Fn(xn), hence I(O!,lcS!,lcBnt)(xn)I - 1 (k - 00). By (1), we have that (O~/lcS!'lcBnt)(Xj) = 0 for j '" n. Hence we can consider O!,Ic S!,Ic Bnt instead of Fn, and so that we may assume moreover that

On < 1 - 1/";1 + 2En.

(4) Let

Then bn(O)

bn (Z) -_

(1 - on) , 1- (1- on)z Z -

zED.

= -1 + On and bn(Fn(xn» = 1- On. Let hn(z) = bn 0 Fn(z)/(I - on),

zED.

Then h n E Hoo,

1Ih,.1I

(5) (6)

h,.(Xn)

= 1,

~ ";1

+ 2En

and h.. (x,,)

= -1

by (4), and for every k with k ::/: n.

lzuchi

226 Now let Then In and gn are contained in Hoo, and by (5)

Moreover by (6),

(7) (8) Here let Gn

= Inglg2 ... gn-l.

Then by [1, Lemma 2] and (3), onD.

(9)

Now we show that {x n },. is interpolating. Let {an}n be an arbitrary bounded sequence. Define By (9), we have G E Hoo, and

by (7) and (8). Here

~:=Io+l an/ nglo+lglc+2

... gn-l is a function in HOO and

(n=le+l 1: anGn){xlc)

=

(glg2··· gle)(Xle) (

=

0

1:

n=le+l

an/ ngle+1··· gn-l)(Xle)

by (8).

Hence G(Xle) = ale for every k. This implies that {xn}n is interpolating. The converse is already mentioned as Fact 2. In the first part of the above proof, we show that if {xn}n is a sequence of trivial points, then there exists a sequence {Fn}n in Hoo such that if k =F n, and IFn(Xn)1 closes to 1 sufficiently for every n.

In the rest of the above proof, we prove that for a given sequence {xn}n in M(HOO), Xn needs not to be trivial, if there is {F.. },. in Hoo satisfying the above conditions, then {x n},. is interpolating. We note that this proof works for the spaces HOO Oil the other domains.

227

Izuchi

PROPOSITION 2. Let HOO be the space of bounded analytic functions on the domain n in cn. Let {xn}n be a sequence in M(HOO). If for every f with 0 < f < 1 there is a sequence {fn}n in HOO such that IIfnll ::;; 1, fn(xk) = 0 for k ::/= nand Ifn(xn)1 > f for every n, then {Xn}n is an interpolating sequence. We have the following problem. PROBLEM 1. Let {xn}n be a sequence of trivial points. (1)

If {Xn}n is strongly discrete, is it interpolating?

(2)

If {xn}n satisfies condition (A 2 ), does it satisfy (A3) ?

We note that Hoffman proved in his unpublished note that if {Xn}n is a strongly discrete sequence in M(LOO), then {xn}n is interpolating.

3. CONDITION (A3) In this section, we prove the following theorem.

THEOREM 2. Let {xn}n be a sequence in G. Then {xn}n satisfies condition (A3) if and only if {xn}n is strongly discrete and there exists an interpolating Blaschke product b such that {xn}n C Z(b). To prove this, we need some lemmas. For a subset E of M(Hoo), we denote by cl E the closure of E in M(HOO).

LEMMA 2 [5, p. 205]' Let b be an interpolating Blaschke product with zeros {zn}n. Then Z(b) = cl {zn}n. LEMMA 3 [6, p. 101]. If b is an interpolating Blaschke product, then Z(b) C G. Conversely, for a point x in G there is an interpolating Blaschke product b such that x E Z( b). For an interpolating Blaschke product b with zeros {zn}n, put

6(b) = inf k

II

n:n~k

p(zn' Zk).

LEMMA 4 [6, p. 82]. Let b be an interpolating Blaschke product and let x be a !,oint in M(Hoo) with b(x) = o. Then for 0 < (1 < 1 there is a Blaschke subproduct B of b such that B(x) = 0 and 6(B) > (1. LEMMA 5 [3, p. 287]' For 0 < 6 < 1, there exists a positive constant K(6) satisfying the following condition; let b be an interpolating Blaschke product with zeros {zn}n such that 6(b) > o. Then for every sequence {an}n of complex numbers with lanl S 1, there is a function h in Hoo such that h(zn) = an for all nand IIhll < K(o).

228

lzuchi

PROOF OF THEOREM 2. First suppose that {xn}n is strongly discrete and {Xn}n C Z(b) for some interpolating Blaschke product b. Take a sequence {Un}n of disjoint open subsets of M(Hoo) such that Xn E Un for every n. Let {zkh be the zeros of b in D. For each n, let bn be the Blaschke product with zeros {zkh n Un. Since {Un}n is a sequence of disjoint subsets, IT::'=lbn is a subproduct of b. By Lemma 2, Xn E cl {zkh. Hence Xn E cl ({ zkh n Un), so that bn(xn) = O. We also have

IeIT

J:J¢n

for every Z E {Zdk

bj)(z)1 2: inf )1. p(Zj, Zi) •

J:J¢'

= 6(b)

> 0

n Un. Hence

satisfies condition (A3)' Next suppose that {xn}n C G and {xn}n satisfies condition (A3)' Then there is a sequence {In}n in HOO such that Ilfnll ~ 1, IT::'=l fn E Hoo, and

Thus

{Xn}n

( 1) for some 6 > O. By considering {cnfn}n with 0 < that

(2)

Ilfnll


> =

,,:!I# If,,(Wk,j) -

g,,(wk,.i)1

II Of,,(Wkj)lK(o) E,,) '

,,:,,~k

exp{ E log(lf,,(Wk,.i)I-K(o)E,,)} ,,:,,~k

by (13) by (12)

lzuchi

230

> exp {

=

c (If,.(wle,j)I- K(c5) E,. - I)}

I;

,.:,.~Ie

exp[-c(

> exp[-c

by (10)

I;

l-lf,.(wle,i)1)] exp(-cK(c5)

I;

-log If,,(wle,j) 11 exp(-cK(c5)c5)

n:n~1e

,.:,.~Ie

I;

,.:,.~Ie

En) by (4) and (11)

( II If,,(wle,j)IYexp(-cK(c5)c5)

=

,.:,.~Ie

> c5 exp(-c K(c5) c5)

by (3) and (7).

C

Therefore we have

c5(b)

=

inf

=

inf Ie,j

Ie"

. p(W,.,;,WIe,j)

. II

(,.,.):(n,.)~(Ie,,)

I(

II

b )(Wle";:;~i ')1 II p(WIe',Il Wle oJ.)

,.:,.~Ie"

> c5c exp (-c K(c5) c5) inf c5(ble) Ie > c5c+1exp(-cK(c5)c5)

by (8).

Thus b is an interpolating Blaschke product. Since bn(xn) = 0, b(xn) = (11:=1 b,.}{x,.) = This completes the proof.

o.

In [8], the author actually proved the following. PROPOSITION 3. Let {xn}n be a sequence in G such that P(X,.) ncl {xleh~,. = 0 for every n. If {x n },. satisfies condition (A 2 ), then {x n},. satisfies condition (A3). If {X,.}n satisfies a more stronger topological condition, then we can get the same conclusion without condition(A2).

PROPOSITION 4. Let {X,.},. be a sequence in G. Ifcl p(x,.)ncl (U1c:1c~n P(XIe» = 0 for every n, then {xn}n satisfies condition (A3). To prove this, we use the following lemma.

LEMMA 6 [8, Lemma 8]. Let x E G and let E be a closed subset of M(H"") with P(x) n E = 0. Then for 0 < E < 1, there is an interpolating Blaschke product b such that b(x) = 0 and Ibl > E on E. PROOF OF PROPOSITION 4. By our assumption, there is a sequence {Un},. . of disjoint open subsets of M(HOO) such that P(X,.) C Un for every n. Let {E,.},. be a sequence of positive numbers such that 0 < E,. < 1 and 11:=1 E,. > o. By Lemma 6, there is an interpolating Blaschke product b,. such that b,.(x,.) = 0 and Ib,.1 > E,. on M(HOO) \ U,.. By considering tails of b,., n = 1,2, ... , we may assume that 11:=1 b,. E Hoo. Since U,. C M(Hoo) \ Ule for k :f:. n, we have

I

II

1e:1e~",

I

ble >

II

1e:1e~,.

Ele

on D

n UrI for every n.

231

Izuchi

Since Xn is contained in cl (D nUn), for every n. Hence {xn}n satisfies condition (A3). In [10], Lingenberg proved that if E is a closed subset of M(Hoo) such that E C G and Hr; = C(E), the space of continuous functions on E, then there is an interpolating Blaschke product b such that E C Z(b). Here we have the following problem. PRPOBLEM 2. If {xn}n is an interpolating sequence in G, is cl {xn}n C G true?

If the answer of this problem is affirmative, we have that if {xn}n is interpolating and {xn}n C G, then there exists an interpolating Blaschke product b such that {xn}n C Z(b). We have anothor problem relating to Problem 2. PROBLEM 3. Let {xn}n be a sequence in G. If {xn}n satisfies condition (A2)' does {xn}n satisfy condition (A3) ? By Theorem 2 and Lemma 3, it is not difficult to see that if Problem 3 is true then Problem 2 is true. We end this section with the following problem. PROBLEM 4. Let {xn}n be a sequence in G. If {xn}n satisfies condition (A2)' is {Xn}n interpolating?

4. CONDITON (Ad In [6, p. 109], Hoffman gave an example of a nontrivial part which is not a homeomorphic part. We use his example to prove the following theorem.

THEOREM 3. There exists a sequence {xn}n satisfing the following conditions.

(i) (ii)

(iii) (iv)

{xn}n {xn}n {xn}n {xn}n

is contained in a nontrivial part. is strongly discrete. satisfies condition (Ad. is not interpolating.

PROOF. We work on in the right half plane C+. Then S = {I + ni}n is an interpolating sequence for HOO(C+). Let b be an interpolating Blaschke product with these zeros. Let the integers operate on S by translation vertically. That gives a group homeomorphism of cl Sj hie : cl S --+ cl S

hle(1+ni) = 1+(n+k)i.

lzuchi

232

Let K be a closed subset of cl S \ S which is invariant under hi and which is minimal with that property (among closed sets). Let mE K. The sequence

k = 1,2, ... is invariant under hi. Therefore for every N.

(1)

Let Lm be the Hoffman map from C+ onto P(m). Then Lm(1) = m and Lm(1 + ik) = m/c. Hence by (1), P(m) is not a homeomorphic part. Let x" = Lm(1+ 1/n+in) for n = 1,2, .... Then

(2)

x" E P(m)

for every n.

We note that

p(l + lIn + in, 1 + in)

(3)

--+

0 (n

-+

00).

Since {I + in}" is an interpolating sequence in C+, {I + lIn + in}" is also interpolating. Hence {I + lIn + in}" satisfies condition(Al). Since Lm preserves p-distance [6, p. 103],

{x"},, satisfies condition (Ad.

(4) Since b = 0 on K, b(x,,)

-+

0 (n

(5)

-+

00). But we have b(x,,)

I- o.

Hence

{x"},, is strongly discrete.

To prove that {x"},, is not interpolating, it is sufficient to prove that {x"},, does not satisfy (A2). Suppose that there exists g" in HOO such that IIg,,1I ~ 1, g,,(x/c) = 0 for k I- n, and g,,(x,,) I- o. By (3), we have p(m/c, Xlc) -+ 0 (k -+ 00). Hence by (1), g" = 0 on K. Therefore

This implies that {x"},, does not satisfy condition (A2).

REFERENCES [1] S. Axler and P. Gorkin, Sequences in the maximal ideal space of Hoo, Proc. Amer. Math. Soc. 108(1990),731-740.

[2] L. Carleson, An interpolation problem for bounded analytic functions, Amer. J. Math. 80(1958), 921-930.

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233

[3] J. Garnett, Bounded analytic functions, Academic Press, New York and London, 1981. [4] P. Gorkin, H. -M. Lingenberg and R. Mortini, Homeomorphic disks in the spectrum of Hoo, Indiana Univ. Math. J. 39(1990), 961-983. [5] K. Hoffman, Banach spaces of analytic functions, Prentice Hall, Englewood Cliffs, New Jersey, 1962. [6] K. Hoffman, Bounded analytic functions and Gleason parts, Ann. 86(1967), 74-111.

of Math.

[7] K. Izuchi, Interpolating sequences in a homeomorphic part of Hoo, Proc. Amer. Math. Soc. 111(1991), 1057-1065. [8] K. Izuchi, Interpolating sequences in the maximal ideal space of Hoo, J. Math. Soc. Japan 43(1991),721-731. [9] K. Izuchi, Factorization of Blaschke products, to appear in Michigan Math. J. [10] H. -M. Lingenberg, Interpolation sets in the maximal ideal space of Hoo, Michigan Math. J. 39(1992), 53-63. Department of Mathematics Kanagawa University Yokohama 221, JAPAM MSC 1991: Primary 30D50, 46J15

234

Operator Theory: Advances and Applications, Vol. 59 © 1992 Birkhauser Verlag Basel

OPERATOR MATRICES WITH CHORDAL INVERSE PATTERNS* Charles R. Johnson 1 and Michael Lundquist We consider invertible operator matrices whose conformally partitioned intler,e, have 0 blocks in positions corresponding to a chordal graph. In this event, we describe a) block entry formulae that express certain blocks (in particular, those corresponding to 0 blocks in the inverse) in terms of others, under a regularity condition, and b) in the Hermitian case, a formula for the inertia in terms of inertias of certain key blocks. INTRODUCTION For Hilbert spaces j{

j{i, i

= 1,··· , n, let j{ be the Hilbert space defined by

= j{1 $ ... $j{n. Suppose, further, that A : j{ -+ j{ is a linear operator in matrix form,

partitioned as

All

A= [

A21

AnI

in which Ai;:

j{; -+ j{i,

i,j = 1,··· ,n. (We refer to such an A as an operator matrix.)

We assume throughout that A is invertible and that A-I = B = IBi;] is partitioned conformably. We are interested in the situation in which some of the blocks Bi; happen to be zero. In this event we present (1) some relations among blocks of A (under a further regularity condition) and (2) a formula for the inertia of A, in teqns of that of certain principal submatrices, when A is Hermitian. For this purpose we define an undirected graph G

= G(B) on vertex set N

== {I,··· , n} as follows: there is an edge {i,j}, if=. j, in

G(B) unless both Bij and Bji are

o.

An undirected graph G is called chordal if no subgraph induced by 4 or more vertices is a cycle. Note that if G(B) is not complete, then there are chordal graphs *Thil manulcript

Wlloll

prepared while both authors were visitor. at the Institute for Mathematics and

its Applications, Minneapolis, Minnesota. lThe work of this author was supported by National Science Foundation grant DMS90-00839 and by Office of Naval Research contract NOOO14-90-J-1739.

Johnson and Lundquist

235

G (that are also not complete) such that G(B) is contained in G. Thus, if there is any symmetric sparsity in B, our results will apply (perhaps by ignoring the fact that some blocks are 0), even if G(B) is not chordal. A clique in an undirected graph G is a set of vertices whose vertex induced subgraph in G is complete (i.e. contains all possible edges {i,j},i '" j). A clique is

maximal if it is not a proper subset of any other clique. Let

e = e( G) = {a1,···

, a p } be

the collection of maximal cliques of the graph G. The intersection graph 9 of the maximal cliques is an undirected graph with vertex set ai

e and an edge between ai and aj, i '" j

if

n aj ",,p. The graph G is connected and chordal if and only if 9 has a spanning tree ai n aj ~ ak whenever ak lies on the unique

T that satisfies the intersection property:

simple path in T from ai to aj. Such a tree T is called a clique tree for G and is generally not unique [2]. (See [3] for general background facts about chordal graphs.) Clique trees constitute an important tool for understanding the structure of a chordal graph. For example, for a pair of nonadjacent vertices u, v in G, a u, v separator is a set of vertices of G whose removal (along with all edges incident with them) leaves u and v in different connected components of the result. A u, v separator is called minimal if no proper subset of it is a u, v separator. A set of vertices is called a minimal vertex separator if it is a minimal u, v separator for some pair of vertices u, v. (Note that it is possible for a proper ,subset of a minimal vertex separator to also be a minimal vertex separator.) If ai and aj are adjacent cliques in a clique tree for a chordal graph G then ai n aj is a minimal vertex separator for G. The collection of such intersections (including multiplicities) turns out

to be independent of the clique tree and all minimal vertex separators for G occur among such intersections. Given an n-by-n operator matrix A = (Aij), we denote the operator subma~rix lying in block rows a ~ N and block columns

Is principal (i.e. fl

= a),

fl

~ N by A[a, fl]. When the submatrix

we abbreviate A[a, a] to A[a].

We define the inertia of an Hermitian operator B on a Hilbert space X as (Qllows. The triple i(B) = (i+(B),i_(B),io(B» ha.') components defined by

i+(B) == the maximum dimension of an invariant subspace of B on which the quadratic form is positive.

L(B) == the maximum dimension of an invariant subspace of B on which the quadratic form is negative. lI.nd

236

Johnson and Lundquist

io(B) == the dimension of the kernel of B (ker B). Each component of i( B) may be a nonnegative integer, or 00 in case the relevant dimension is not finite. We say that two Hermitian operators Bl and B2 on X are congruent if there is an invertible operator C : X

-t

X such that

According to the spectral theorem, if a bounded linear operator A : 1{ - t 1{ is Hermitian, then A is unitarily congruent (similar) to a direct sum:

o A_

o in which A+ is positive definite and A_ is negative definite. As i(A) = i(U* AU), i+(A) is the "dimension" of the direct summand A+, L(A) the dimension of A_, and io(A) the dimension of the 0 direct summand, including the possibility of 00 in each case. It is easily checked that the following three statements are then equivalent:

.

(i) A is congruent to

[I0 o

0 0]

-I 0

0 , in which the sizes of the diagonal blocks are

0

i+(A), L(A) and io(A), respectively; (ii) each of A+ and A_ is invertible; and (iii) A has closed range. We shall frequently need to make use of congruential representations of the form (i) and, so, assume throughout that each key principal submatrix (i.e. those corresponding to maximal cliques and minimal separators in the chordal graph G of the inverse of an invertible Hermitian matrix) has closed range. This may be a stronger assumption than is necessary for our formulae in section 3; so there is an open question here. Chordal graphs have played a key role in the theory of positive definite completions of matrices and in determinantal formulae. For example, in [4] it was shown that if the undirected graph of the specified entries of a partial positive definite matrix (with specified diagonal) is chordal, then a positive definite completion exists. (See e.g. [6] for definitions and background.) Furthermore, if the graph of the specified entries is not chordal, then there is a partial positive definite matrix for which there is no positive

Johnson and Lundquist

237

definite completion. (These facts carry over in a natural way to operator matrices.) If there is a positive definite completion, then there is a unique determinant maximizing one that is characterized by having O's in the inver.,e in all positions corresponding to originally unspecified entries. Thus, if the graph of the specified entries is chordal, then the ordinary (undirected) graph of the inverse of the determinant maximizer is (generically) the same chordal graph. (In the partial positive definite operator matrix case such a zeros in-theinverse completion still exists when the data is chordal and is an open question otherwise.) This was one of the initial motivations for studying matrices with chordal inverse (nonzero) patterns. Other motivation includes the structure of inverses of banded matrices, and this is background for section 2. If an invertible matrix A has an inverse pattern contained in a chordal graph G, then det A may be expressed in terms of certain key principal minors [1], as long as all

relevant minors are nonzero:

IT det A[a] detA= __~o~E_e____~__~ IT det A[a n .0]. {o,P}E£

Here

e is

the collections of maximal cliques of G, and 'J =

(e, e)

is a clique tree for G.

Thus, the numerator is the product of principal minors associated with maximal cliques, while the denominator has those associated with minimal vertex separators (with proper multiplicities). There is no natural analog of this determinantal formula in the operator case, but the inertia formula presented in section 3 has a logarithmic resemblance to it. 2. ENTRY FORMULAE

Let G = (N, E) be a chordal graph. We will say that an operator matrix

A

= [Ai;] is G-regular if A[a] is invertible whenever

0'

~

V is either a maximal clique of G

or a minimal vertex separator of G. In this section we will establish explicit formulae for some of the block entries of A when G(A -I)

~ G.

Specifically, those entries are the ones

corresponding to edges that are ab.,ent from E (see Theorem 3). LEMMA

1. Let A = [Aij] be a 9-by-9 operator matrix, and assume that

A12] A22 '

M2

=

[~:~ ~::]

and

A22

are each invertible.

238

Johnson and Lundquist

Proof. Let us compute the Schur complement of A22 in A:

[~

(1) =

If B

-AI2A2"1 I -A32A21

[ An -

;]

A"A,,' A"

A31

Al2 A22 A32

A 23 A33

0

Au -

AnA;,' A" ]

[An A21

0

A22

A31 - A32A21 A21

0

AU] [ - A221I A21 o

0

I 0

0

-~A" ]

.

A33 - A32 A221 A23

= A-I exists, then

(2)

[

Bll B31

and hence if Bl3

= 0, then necessarily we have A13 = A12A221 A 23 .

Conversely, if A13

=

Al2A221 An, then the (1,3) entry of the matrix on the right-hand side of (1) is zero. Note that All -- Al2A21 A21 and A33 - A32A221 A 23 are invertible, because they are the Schur complements of A22 in MI and M 2. Hence A is invertible, and by (2) we have B13

= o.

0

Under the conditions of the preceding lemma, if we would like B31 = 0 then we must also have A31 = A32A221 A21 . Notice that the graph of B in this case is a path:

G

= m.

In this case B has the block form

in which Bll = B[{I, ... ,k -I}], B22 = B[{k, ... ,m}] and B33 = B[{m + 1, ... ,m}]. Let A = [Aij] be partitioned conformably. If in addition to the above conditions we also have

that A[{I, ... , m}], A[{k, ... , n}] and A[{k, ... , m}] are invertible, then we simply have the case covered in the preceding Lemma, and we may deduce that

A[{I, ... ,k - I}, {m + 1, ... ,n}] = A[{I, ... , k - I}, {k, ... , m}] A[{k, ... ,m}]-l A[{k, ... , m}, {m + 1, ... ,n}],

with a similar formula holding for A[{m

+ 1, ... , n}, {I, ... , k - I}]. From this we may

write explicit formulae for individual entries in A. For example, we may express any entry

Johnson and Lundquist

Aij

239

for which i < k and j > m as

(3)

Aij

= A[{i}, {k, ... , m}] A[{k, ... , m}]-I A[{k, ... , m}, {ill.

There is an obvious similarity between this situation and that covered in Lemma 1, which one sees simply by looking at the block structure of A -I. But there are also some similarities which may be observed by looking at graphs. In the block case we just considered, the graph G(B) is a chordal graph consisting of exactly two maximal cliques, the sets 01

= {l, ... ,m} and 02 = {k, ... , n}.

The intersection

13 = {k, ... , m}

of 01 and a2 is a

minimal vertex separator of G (in fact, the only minimal vertex separator in this graph). The formula (3) may then be written

(4)

Aij

= A[{i}, f3]A[f3r l A[f3, {j}].

Note now in the 3-by-3 case that the equation when we let

13 = {2}.

A13

= A12A221 A23 has the same form as (4)

In fact, since 13 = {2} is a minimal separator of the vertices 1 and 3

in the graph

we see that {2} plays the same role in the 3-by-3 case as {k, ... , m} does in the n x n case. In Theorem 3 we will encounter expressions of the form

in which each 13,. is a minimal vertex separator in a chordal graph. The sequence (131 , ... ,13m)

;is obtained by looking at a clique tree for the chordal graph, identifying a path (aD, al, ... , am) in the tree, and setting

13,. =

a"_1

n ak.

These expressions turn out to be the natural generalization of (4) to cases

in which the graph of B is any chordal graph. In addition, the results of this section ,generalize results of [9] from the scalar case to the operator case. LEMMA 2.

Let A : 1( -+ 1( be an invertible operator matrix, with B

Let G = (N,E) be the undirected graph of B. Let {i,j} fj. E and let 13 separator for which A[f3] is invertible. Then Aij

= A[{i},f3]A[.B]-IA[.B,{i}].

~

= A-I.

N be any i,j

Johnson and Lundquist

240

Proof. Without loss of generality we may assume that fJ = {k, ... , m}, with

k :5 m, and that fJ separates any vertices r and s for which r < k and s > m. Assuming then that i < k and j > m, we may write B as

~12

0

l!.22

~23

B22

B33

1

•

The result now follows from Lemma 1 and the remarks that follow it.

0

If G is chordal and i and j are nonadjacent vertices then an i, j clique path will mean a path in any clique tree associated with G that joins a clique containing

vertex i to a clique containing vertex j. One important property of any i,j clique path is that it will "contain" every minimal i,j separator in the following sense: If (0'0, ... ,am) is any i,j clique path, and if fJ is any minimal i,j separator then fJ = ak-l n ak for some k,l :5 k :5 m. Another important property of an i,j clique path is that every set fJle = ale-l nale, 1 :5 k:5 m, is an i,j separator. It is not the case, however, that every fJle

is a minimal i,j separator (see [9]). THEOREM

1( be

3. Let G = (N, E) be a connected chordal graph, and let A: 1(--+

a G-regular operator matrix. then the following assertions are equivalent:

(i) A is invertible and G(A-l) ~ G; (ii) for every {i,j} ¢ E there exists a minimal i,j separator fJ such that Ai; = A[{i},fJl A[fJl- 1 A[fJ, {ill;

(iii) for every {i,j} ¢ E and every minimal i,j separator fJ we have Ai; = A[{i},fJl A[fJl- 1 A[fJ, {j}I;

(iv) for every {i,j} ¢ E, every i,j clique path (ao,aJ, ... ,a m) and any k,l:5 k:5 m we have

in which fJle = ale-l

n ak;

and

(v) for every {i,j} ¢ E and every i,j clique path (0'0, aJ, ... am) we have

in which fJle = ale-l n ale.

241

Johnson and Lundquist

Proof. We will establish the following implications:

(iv) ==> (iii) ==> (ii) ==> (iv)j (i) {:::::} (iv) {:::::} (v).

equals

p,.

(iv) ==> (iii) follows from the observation that every minimal i,j separator for some k, 1 ~ k ~ m. (iii) ==> (ii) is immediate. For (ii) ==> (iv), let {i,n ¢ E, and let (ao,a}, ... ,a m ) be a shortest i,j

clique path. We will induct on m. For m = 1 there is nothing to show, since in this case PI = ao

n aI

is the only minimal i,j separator. Now let m

~

2, and suppose that (iv)

holds for all nonadjacent pairs of vertices for which the shortest clique path has length less than m. Since every minimal i,j separator equals

for some k, 1

~

k

~

p", for some k, we have, by (ii),

m. It will therefore suffice to show that for k = 1,2, ... , m - 1 we

have

Let us first observe that for k = 1, ... , m - 1,

(7) Indeed, suppose rEP",. Then (a,., ak+}' ... , am) is an r,j clique path of length m - k, and by the induction hypothesis we may write

and equation (7) follows. A similar argument shows that for k = 2, ... , m we have

(8) By (7) and (8), both sides of (6) are equal to

Johnson and Lundquist

242

and hence (6) holds, as required.

(i)

===?

(iv) follows from Lemma 2.

For (iv)

(i), let the maximal cliques of G be aI,a2, ... ,a" P

===?

~

2. We

will induct on p. In case p = 2 then the result follows from Lemma 1, so let p > 2 and suppose that the implication holds whenever the maximal cliques number fewer than p. Let 'J be a clique tree associated with G, let {ak,ak+d be any edge of 'J, and suppose the vertex sets of the two connected components of 'J - {ak,ak+d are el = {al,'" ,ak} and

e2 = {ak+b ... ,a,}.

Set

Vi = U~=lai and

V2

= Uf=k+lai.

(Let Gv be the subgraphof

G induced by the subset V of vertices.) Since induced subgraphs of a chordal graph are necessarily chordal, Gv, and GV2 are chordal graphs, and since (iv) holds for the matrix A, (iv) holds as well for A[Vd and A[V2]. By the induction hypothesis, A[VI] and A[V2]

are invertible. Note also that Vi n V2 = ak n ak+h which follows from the intersection property. Since A[VI n V2] is invertible, we may now apply Lemma 1 to the matrix A (in which Au is replaced by A[VI \ V2], A22 by A[VI nV2] and A33 by A[V2 \ Vi]), and conclude that A-I[VI \ V2, V2 \ VI] = 0 and A- I [V2 \ V}, VI \ V2 ] B = A-I then Bij = 0 and Bji = 0 whenever i E

=

O. In other words, if we set

Vi \ V2 and j E V2 \

VI. Now if {i,j}

rt E

then ak and ak+I may be chosen (renumbering the a's if necessary) so that i E VI \ V2 and j E V2 \ VI. Hence it must be that Bij = 0 and Bji = 0 whenever {i,j} For (iv)

===?

(v), let {i,j}

rt E, and let (ao, aI, ... , am) be any i,j

path. First, we must observe that for any k, 1 ::; k ::; m,

(9) Indeed, by assumption, for any r E f3k we have

and (9) follows from this. By successively applying (9) we obtain Aij = A[{i},f3I]A[f3I]-1 A[.BI, {j}]

= A[ {i}, .BdA[.BI]-1 A[.BI, .B2]A[.B2]-1 A[.B2, {j}]

as required.

rt E. clique

Johnson and Lundquist

For (v) Let r E

243

:=}

0'''-1, 1 < k :::; m.

and because rEfit

:=}

(iv), let {i,n ¢ E, and let (0'0, ... , am) be an i,j clique path.

We may write [because of assumption (v)]

rEa" we thus have

(10)

It may be similarly shown that (11) By using (10) and (11) we therefore obtain Aij

= A[{i},.81]··· A[.8k-t. .8,,]A[.8k]-1 A[.8k,.8k+d··· A[.8m, {j}] = A[{i},.8,,]A[.8k]-1 A[.8", {ill,

lIS

required.

0

3. INERTIA FORMULA In [8], it was shown that if A E Mn(C) is an invertible Hermitian matrix and if G = G(A-l) is a chordal graph, then the inertia of A may be expressed in terms of the inertias of certain principal submatrices of A. Precisely, let

e denote the collection of

maximal cliques ofG, and let 'J" = (e,e) be a clique tree associated with G. IfG(A- 1 ) = G, then it turns out that (11)

i(A)

=

L

i(A[a]) -

arEe

L

i(A[a

n .8]).

{ar,p}E£

:.It is helpful to think of (11) as a generalization of the fact that if A -1 is block diagonal (meaning, of course, that A is block diagonal) then the inertia of A is simply the sum of the inertias of the diagonal blocks of A. To see what (11) tells us in a specific case, suppose :that A-I has a pentadiagonal nonzero-pattern, as in X

A-I ~

[

X

X

XiiX

X X

X

X

~

i1

X X

Johnson and Lundquist

244 The graph of A-I is then

G=~ ~

which is chordal. The maximal cliques of G are

0'1

= {I, 2, 3},

0'2

= {2, 3, 4} and

0'3

=

{3,4,5}, and the clique tree associated with the graph G is

®---&--§). Equation (11) now tells us that the inertia of A is given by i(A) = i(A[{1,2,3}])

+ i(A[{2,3,4}]) + i(A[{3,4,5}])

- i(A[{2, 3}]) - i(A[{3, 4}]).

Thus, we may compute the inertia of A by adding the inertia of these submatrices:

x X X X X

[

X X X X X

X X X X X

X X X X X

X X X X X

]

and subtracting the inertias of these:

Our goal in this section is to generalize formula (11) to the case in which A = [Aiil is an invertible n-by-n Hermitian operator matrix. We will be concerned with

the case in which one of the components of inertia is finite, so that in (11) we will replace i by i+, L or i o •

For a chordal graph G = (N, E), we will say that an invertible n-by-n operator matrix A is weakly G-regular (or simply weakly regular) if for every maximal clique or minimal vertex separator a both A[a] and A-I [a C ] have closed range.

Johnson and Lundquist

LEMMA

245

4. Let M:

1(1 EB1(2 --+ 1(1 EB1(2

be represented by the !-by-2 matrix

M=[~ ~]. Suppose that A is invertible, and that

[~ ~].

M- 1 =

Then dim ker A

= dim ker s.

Proof Let

Xl, X2, ••• ,X n

be linearly independent elements of ker A. Then

for 1 :::; k :::; n we have

eXIe,

in which Yle =

k = 1, ... , n. Since M is invertible it follows that Yb Y2, ... , Yn are

linearly independent. Observe now that

from which it follows that Yle E ker s. It follows now that dimker S versing the argument we find that dim ker A LEMMA

5. Let M : 1(1

~

~

dimker Aj by re-

dim ker S. Thus dim ker A = dim ker S.

EB 1(2 --+ 1(1 EB 1(2

0

be Hermitian and invertible, and

suppose that

If i+(M)
1,8('1') = {ab.'. ,an}, -00 < al < ... < < an < 00, and let ti, i = 1, ... , n -1, be real points with ai < ti < ai+1, i = 1, ... , n -1, such that 'I' has no masses in the points tie We set ~1 := (-00, tl], ~i := (ti-b til, i = 2, ... , n -1, ~n := (tn-I, 00). A system ofintervals ~i' i = 1, ... ,n, with these properties is called a cp-minimal decomposition of m. Let XLl., be the characteristic function on m of ~i' Then XLl.,cp E :F (m, ai), i = 1, ... , n, and n

'1'. / = L(XLl.icp)· /,

/ E()OO (m).

i=l

Here"·" denotes the usual duality of distributions and ()OO functions on m. If a E m and 'I' E :F (m, a), the order I' ('I') of 'I' is, as usual, the smallest n E 1No (= 1N U {O}) such that 'I' is the n-th derivative of a (signed) measure on m. We denote by 1'0 (aj '1'), (I'r (aj '1'), 1'1 (aj '1'» the smallest n E 1No such that, for some measure T'

254

Jonas et al.

on IR, the n-th derivative T(n) OfT and cP coincide on (-oo,a) U (a,oo) «a,oo), (-oo,a), respectively). The numbers /-Lo (aj cp), /-Lr (aj cp), /-LI (aj cp) are called reduced order, right reduced order, left reduced order, respectively, of cp at a. Evidently, we have /-Lo (aj cp) = max {/-Lr (aj cp), /-LI (aj cP Some more properties of these numbers are given in the following lemma (compare [6j Hilfsatz 1,2]).

n.

LEMMA 1.1. If cp E F(IR,a), then the following statements hold. (i) /-Lr (aj cp) (/-LI (aj cP » coincides with the minimum of the numbers n E IN 0 such that (t - a)ncp is a bounded measure on (a, 00) ( (-00, a), respectively).

(ii) (t - a)I'('I')cp is a measure. Here and in the sequel t denotes the function / (t) == t.

PROOF. (i) Choose a> lal with supp cp c (-a, a) and let n be such that (t-a)ncp is a bounded measure on (a,a). If / is an element of

M:= {f

E

C.;"'(IR) : supp / C (a,a),

sup IJCn)(t)l-:::; I}, tE (a,a)

Taylor's formula implies sup{l(t - a) sup{lcp·

/1: / E M}

-n /

(t)1 : t

E (a,

an -: :; 1. It follows that

= sup{l(t - a)ncp. (t - a)-n/I: / E M}

< 00.

Then, by a standard argument of distribution theory, cp restricted to (a,a) is the n-th derivative of a bounded measure on (a, a), i.e. /-Lr (a j cp) -:::; n. It remains to show that with /-Lr := /-Lr (aj cp), (t - a) I'F cp I(a, a) is a bounded measure. To this end choose a nonnegative f3 E Coo (IR) equal to on a neighbourhood of (-00,0] in IR and equal to 1 on a neighbourhood of [1,00) in IR, and set f3dt) := f3 (k (t - a) ), k E IN. A simple computation yields the uniform boundedness of the functions a)I'Ff3k)(I'F),k E IN, on (a,a). Then, since there exists a measure CPo on IR with compact support such that cp~I'F) = cp on (a,oo), it follows that

°

«t -

(t - a )I'F cp . f3k = cp . (t - a )I'F f3k = (-1 )I'F CPo • ( (t - a}l'F f3k) (I'F). The last expression is uniformly bounded with respect to k, hence (t - a)I'Fcp is a bounded measure on (a,a). The claim about /-LI (ajcp) is proved analogously.

(ii) Evidently, /-L := /-L (cp) ;::: /-Lo (aj cp). Then by (i) there exists a measure CPo on IR with supp CPo C (-a, a) such that (t - a) I'cp

= CPo +

L• aID~), 1=1

where Da is the D-measure concentrated in the point a. Let I E IN and define /!,.(t) := (t - a + 10)' for t -:::; a - f,/!,.(t) = (t - a - 10)' for t ;::: a + f,/,At) = for t E (a - f,a + f). Then we have

°

(t - a)l'cp . (t - a)' = lim(t - a)l'cp . (t - a)-I' /1+1',' = .-+0

Jonas et al.

It follows that

255

a, = 0,1 = 1, ... , s, and (ii) is proved. .r

1.2. Integral representations of the distributions of (IR). The distributions of the classes (IR, 0) can be represented by certain measures. Consider cp E (IR, 0) and set

.r

k.= {lI'O(Ojcp) .

1(l'o(ojcp) + 1)

.r

if 1'0(Ojcp) is even, if 1'0(Ojcp) is odd.

Then, by Lemma 1.1, the distribution (t - 0)2lecp is the restriction to 1R\{0} of a positive measure u on IR with compact support and u( {o}) = 0. If k ~ 1, then the function (t - 0)-2 is not u-integrable,

f

(t - 0)-2du (t) = 00.

R

Indeed, otherwise (t - 0) 2le-2cp would be a bounded measure on IR\{o}. Then, as 2k - 2 ~ 1'0 ( OJ cp) -1 ~ 0, the distribution (t - 0) 1'0 (Q;'P) -lcp would be a bounded measure on 1R\{ o}, which in view of Lemma 1.1 is a contradiction to the minimality of 1'0 (OJ cp). By the definition of u the distribution (t - o?lecp - u is concentrated in the point o. If k = 0, define Ci:=(CP-U).(t-O)i, i=O,1, ... ,

if k

~

1, ._{cp.(t-O)i c. o)2lecp - u) . (t - 0)i-2le

«t _

By Lemma 1.1, Ci =

fori=0, ... ,2k-1 for i = 2k,2k + 1, ...

°

if i is larger than the order of cpo

In the sequel for a function

f

with n derivatives at t =

0

we use the notation

71-1

f{Q,O}(t):= f(t), f{Q,n}(t):= f(t) -

L i!-l(t -

o)if(i)(o)

i=O

for n ~ 1, or, if 0 is clear from the context, shorter f{O} and f{n}, respectively. Then, for every f E GOO(IR),

Jonas et aI.

256 = u·

(t - a)-2" j{a,2"}

+E

ci

i!-l j E :F is uniquely determined by its restriction to the linear space P of all polynomials. 2. THE PONTRJAGIN SPACE ASSOCIATED WITH 4> E:F. 2.1. Completions. Let (.c, [.,.]) be an inner product space (see [3]), that is, .c is a linear space equipped with the hermitian sesquilinear form h .J. By K_ (.c) we denote the number of negative squares of.c or of the inner product [., .J, that is the supremum of the dimensions of all negative subspaces of .c (for the notations see also [3J, [2]). In what follows this number is always finite. Further,.c° := {z : [z,.cJ = {On is the isotropic subspace of (.c,[.,.]), and it is well-known that the factor space (.c/.c 0 ,[.,.]) admits a unique completion to a Pontrjagin space ([5, 2]), which is denoted by (.c/ .cO )-. LEMMA 2.1. Let .c be a linear space, which is equipped with a nonnegative inner product (.,.), and suppose that on .c there are given linear functionals VI, ••• , Vn such that no linear combination of the Vj' 8 is bounded with respect to the seminorm (.,. )l. Further, . let A = (Ojll)1' be a hermitian n x n-matrix which is nonsingular and has K negative (and

Jonas et al.

258

n - It positive) eigenvalues counted according to their multiplicities. Consider on C the inner product n

[z,y) = (z,y)

+

L

QjleVk(Z)

vi(y) (z,y E C).

i,k=1

Then this inner product has It negative squares on C. The completion of (Cj Co, [.,.J) is the Pontrjagin space 1£ EB A where 1£ is the Hilbert space completion of £ := Cj Co, Co := ={zEC: (z,z) =O}, andA:=(GJ n , (A·,·)GJ"). More exactly, the mapping

is an isometry of (C, [.,.)) onto a dense subspace of 1£ EB A, ker t = Co. PROOF. Evidently, the mapping t is an isometry'of (C,[·,·)) into the 7r,.-space 1£ EB A. Hence (C, [.,.)) has a finite number (~ It) of negative squares. In order to prove that the range of t is dense in 1£ EB A we show that for each io E {I, ... , n} there exists a sequence (y,,) C C such that (y",y,,) -+ 0, vi(y,,) -+ if i E {I, ... ,n}\{io} and vio (y,,) -+ 1, II -+ 00. Indeed, if for each sequence (y,,) the first two relations would imply vio(y,,) -+ 0, then vio would be a continuous linear functional on C with respect to the semi-norm

°

C :3 Y -+ (y,y)!

+

(t

1

IVj

(y) 12 ) "

'''''0 This would imply a representation n

Vio(Y) = (y,e)

+ LVi (Y)Vj

(y E C),

;=1 ;"';0

with

e E 1£ and Vi

E GJ, which is impossible, as no nontriyial linear combination of the

vi's is a continuous linear functional on (C,h·)l). The sequence (ty,,) converges to

/;0 := (OJ 0, ... ,0, 1,0, ... ,O)T E 1£ EB A, where the 1 is at the io-th component. It follows that for arbitrary z E C the element tZ-Vl(Z)JI- ... -Vn(Z)/n= (ZjO, ... ,O)T is the limit in 1£ EB A of a sequence belonging to t (C). Hence" (C) is dense in 1£ EB A. The proof of the following lemma is straightforward and therefore left to the reader.

Jonas et al.

259

LEMMA 2.2. Let (C, [-,.J) be an inner product space and

be a direct and orthogonal decomposition. Then for the isotropic parts it holds

and

C/C o = CdC~[+]C2/cg[+] ... [+]Cn/C~, If K._ (C) < 00 then a sequence in C/ CO is a Cauchy sequence if and only if for each k = 1,2, ... ,n the corresponding sequence of projections in C,./ C~ is a Cauchy sequence, moreover

2.2. Inner products defined by the functionals of F. Let E F be given and denote by 'P the linear space of all polynomials. On'P an inner product [., .]", is defined by the relation [/,g]", := (fg) (f,g E 'P),

where g (z) := g (z). This inner product can be extended by continuity to linear spaces which contain 'P as a proper subspace. To this end we observe the decomposition

= r.p +.,p, r.p E F(m.), .,p E F(4J\m.), which implies

[/,g]", = [/,g]¥'

+ [/,g]",

(f,g E 'P).

Now [., .]", can be extended by continuity to Coo (m.) x H(lTo ()) or to B2 (r.p) x H (lTo ()), where B2 (r.p) is the linear space of all functions / on m. such that (i) / restricted to m.\s(r.p) is r.p- measurable and

J 1/1 2 dr.p


0, the restriction of / to {t Em.: dist (s (r.p), t) < b/} is a Coo- function.

If s(r.p) f: 0,s(r.p) = {a1, ... ,a n }, let (.6. i )i"=l be a r.p-minimal decomposition of m. (see Section 1.1) such that ai belongs to the interior of .6. i ; if s(r.p) = 0 choose n = 1 and.6. 1 = m.. Further, if.,p f: 0, lTo(.,p) = {f31, ... ,f3m'P1, ... ,Pm}, choose mutually disjoint nei~bourhoods Uj of {f3;.p j },j = 1,2, ... ,m, which do not intersect the real axis, U:= U Uj. Define functions X.t., and Xj as follows: j=l

Xj (z) :=

I { 0

if if

Z Z

E Uj E ( U U,.) "~j

u m..

Jonas et al.

260

If £Ie is the linear subspace of B2 ( PI, Pi. In view of the relation

k-l (pq){2k} (t) = p{k}(t)q{k}(t) + L (t - ali (Piq{2k-j}(t) i=o

+ "iii

p{2k-i }(t»

it follows that

[p,q]."

=

J J

(t - a)-2kp{k}(t)q{k}(t) dO'(t)

R

=

k-l

I

J=O

1=0

+ ~(Pi Qj + "iiiPj) + ~ Ci k-l

(t - a)-2kp{k}(t)q{k}(t) dO'(t)

R

i=O

k-l I + L"iii(L cHiPi .=0 i=k

=

k-l

+ L"iii L

k-l

CHi Pi

i=o

I

+ Pi) + LPi (L cHi"iii + "ii;)P + Qj)

J

i=o

i=k

(t - a)-2k p{k} (t)q{k} (t)dO'(t)

R

+

k-l

k-l

1=0

J=o

?:"iii ~ CHiP;

k-l k-l + L"iiiPi+ LPiQi' i=O i=o where we agree that

C II

= 0 if v> 1.

This relation can be written as

with the Gram operator G as in (2.6). Since (Gz, z) = 0 for all elements

L

.Pp."iill =

1"+11=1

Jonas et aI.

265

:z: = (OjO, ... ,Ojeo, ... ,6,-d T ,

eo, ... ,e1e-l E

{l,

e.g. the minimax characterization of the negative eigenvalues of G implies that G has k negative eigenvalues, counted according to their multiplicities. On the other hand, /top = k if I :$ 2k - 1 ( see (2.8». According to Lemma 2.4, no linear combination of Po, ... ,P1e-l,PO, ••• ,P1e-l is continuous with respect to the seminorm

P

~

(1 (. -

1

a)-" Ip{'}(')

1'.10-('») • ,

and the claims of Theorem 2.5 in case 1:$ 2k -1 follow from Lemma 2.1. Assume now that 1 ;::: 2k. It follows as above that

[p,q)op =

J

(t - a)-21e p{1e}(t)q{1e}(t)do" (t)

R

+

~qi ~ CHi Pi + ~qi (.3=1-1e+l t CHj Pi + PI) .=0

3=0

+~Pj( i=o

~

.=0

t

CHjqi+Qi).

i=I-1e+l

This relation can be written as

[p,q)op = (G£opP,£opq) L2(C7)e{l2Hp

(p,q E 'P)

..nth the Gram operator G in (2.7). The number of negative eigenvalues of G is equal to k + ¥if r is even and, if r is + p + 1 if c, < and k + p if c, > 0. This is just the value ~Ip (see (2.8» in the case I > 2k. Now all assertions of Theorem 2.5 for the case I > 2k follow as above from the Lemmas 2.1 and 2.4.

~d, r = 2p + 1, it is equal to k

°

Evidently, in Theorem 2.5 the linear space P can be replaced by Coo (JR.) or B2 (cp). 2.4. The spaces n(4)). Now let 4> E :F,4> = cp+1/J with cp E :F(JR.),1/J E :F({l\JR.). :The considerations in Subsection 2.2, in particular (2.1), and the fact that all the inner ,product spaces on the right hand side of this relation have a finite number of negative i~quares (see Proposition 2.3 and Theorem 2.5) imply that (B2 (cp) x H (0'0 (4))), h·)",) has "finite number of negative squares. We denote the completion of (B/BO,[.,.)",) where • := B2 (cp) x H (0'0 (4))), by n (4)). On account of Lemma 2.2 this space decomposes as follows:

266

Jonas et al.

Here (~i)f=l is again a ~-minimal decomposition of R and X~l' ... ,X~ .. , Xl, ... ,Xm are defined as in 2.2. Models of the spaces n(X~; ~), n (Xi 1/1 ), i = 1, ... , n,j = 1, ... , m, were given in Proposition 2.3 and Theorem 2.5, and a corresponding model of n (t/J) is a direct orthogonal sum of models of these types. The number of negative squares of n (t/J) is then, evidently, the sum of the numbers of negative squares of all the components on the right hand side of (2.9), which are given by (2.3) and (2.8), respectively. Concluding this section we mention the following. The starting point of the above considerations was an inner product [-, .] .. on the space 'P, given by some t/J E :F, and it turned out that this inner product has a finite number of negative squares. We could have started from any inner product [.,.] on 'P with a finite number" of negative squares. Then, up to a positive measure near 00, the inner product [-,.] is generated by some t/J E :F. Namely, for each sufficiently large bounded open interval ~ there exist a measure U oo on R\~ and a t/J E :F which is zero on R\~ such that

(p,qj = (p,q] .. +

(2.10)

J

p(t)q(t)duoo(t) (p,q E 'P).

R\~

Here the measure u 00 is such that

J

IW'duoo(t)
introduced in (2.5), we write

and express the components of this vector by those of 'v> (p). Evidently,

Po Further,

= 0,

Pi = Pi-I,

j = 1,2, ... , k - 1.

Jonas et al.

269

and we find (with evident notation) Pj =

=

f f

r(21e- j )(tp){2k- j }(t)doo(t) r(2k- j -l)p{21e- j -l}(t)doo(t)

-

= Po

-

/

= CkPk-l

/-1

j=k

j=k-l

/-1

/-1

~

~

-, LJ Cj+1Pj + PI =

, LJ Cj+1Pj-l + P 2 =

j=k

j=k

-

~

+ PI

LJ ;=k-l

+ P 2,

/

=L

Cj+1Pj

1,

+ PI,

/

= Ck+lPk-l

A-I

/

~ -, ~ , ~ = LJ CjPj + Po = LJ Cjpj-l + PI = LJ

j=k

PI =

= Pj+1' j = 0,1, ... , k -

/

CjH-IPj

+ P~-1 = L

~k

CjH-IPj-l

+ P~

~k

/-1

=

L

CjHPj

+ P~ =

C2k-lPk-l

+ Pk

j=k-l

where Pk :=

f

rkp{k}(t)doo(t) = (t-kp{k}, 1)"..

It follows that AlP has the matrix representation (3.1) with a

=

o.

The proof of the following theorem about the matrix representation of the operator 000 (t/J) = {~,/~}, is similar but much simpler and therefore left to the

/A""t/J E .1"(C\lR), te8.der.

THEOREM 3.3. Let t/J E .1"(C\lR), ooo(t/J) = {~,p} and suppose that t/J has the form (2.2). Then in the space C 211 , equipped with the same inner product as in Proposition 2.3, the operator A" admits the matrix representation ~

0

1

o

0 1

o

~

1

o

o

1

13

Jonas et at.

270

3.2. Eigenspace, resolvent and spectral function of A",. In this section, if 'P E .r(JRja), for the model operator A", in L2(0') EaGJ 2 1e+ r from (3.1), (3.2) we study the algebraic eigenspace at a and the behaviour of its resolvent and spectral function near a. Without loss of generality we suppose that a = 0.

°

First we mention that a = is an eigenvalue of A", with a nonpositive eigenvector and that A", has no other eigenvalues with this property. A maximal Jordan chain zo, Z1, ••• , z ... of A", at a = is given as follows:

°

If I :::; 2k - 1, then Til

=k-

1 and Zk-1

= (OjO, •.. ,Ojl,O, ... ,O)T,

Zk-2

= (OjO, ... ,OjO,I, ... O)T,

Zo =

(OJ 0, ... , OJ 0, 0, ... , 1) T

and the span of these elements is neutral. If I ? 2k then m = I - k + 1 and Zl-k

= (OJ 0, .•. , OJ 1,0, ... , OJ 0, ... ,0) T,

ZI-k-1

= (OjO, ... ,OjO,I, ..• ,OjO, .•. ,O)T,

Zk

= (OjO, ... ,OjO,O, ... ,ljO, ••. ,O)T,

Zk-1

Zo

= (OJ 0, ... , OJ 0, ... , OJ Cl, CI-1,. ••

, Cl-k+d T,

= (OjO, •.. ,OjO, ... ,OjO, .•. ,O,CI)T.

In the second case it holds

l-k ([zi,zjD.. Z,J =

°= ["° ° ° Cl

°° C21.+1

Cl

,,L,l C2k

where [-,.J := (G·, ·)L.(O')E!IGJ •• +r (see (2.7». Hence the elements zo, ... ,Zk-1 span a neutral subspace, whereas on the span of Zk, ••• , Zl-k the inner product [-,.J is nondegenerate. Next we consider the matrix representation of the resolvent of A",. For the sake of simplicity we write down the matrix (A", - z1)-1 if k = 2 and I :::; 21: -1 = 3; its structure for arbitrary k and I ~ 21: - 1 will then be clear. This matrix is

Jonas et al.

271 (t - z)-I. 0 0 -Z -1(.,(t-z) -1).,. -z-2(.,(t _ Z)-I).,.

[

Z-I(t - z)-1 0 -Z-I b12(z)

z-2(t - Z)-I -Z -1

_z-2 bl1(Z) b21(z)

~2(Z)

0 0 0 -z 1

_z-2

0 0 0 0 -z -I

J

where

(bij(zm _ [ -csz- s + z-s J(t - z)-ldO'(t) -caz-4 - C2Z-S + Z-4 J(t - z)-ldO'(t)

-csz- 2 + Z-2 J(t - z)-ldO'(t) ] -C3Z-3 - C2Z-2 + Z-3 J(t - z)-ldu(t)

The growth of (Atp - zI)-1 if z approaches zero along the imaginary axis or, more generally, nontangentially, is given by the term ~I(Z). In the general case the growth of (Atp - zI)-1 is also determined by the entry b on the second place of the last row. If I ~ 2k - 1, we have

b ( z ) = -CkZ -k-I - Ck+IZ -k-2 - ... - C2k-IZ -2k and, hence, for z = iy and y

+ z -2k Jet -

Z)-Id0'(t)

1 0,

If 1 ~ 2k, we have

b( z ) =

-CIZ -I-I -

CZ-IZ -I - ... - CI_k+IZ -l+k-2 - Z-2k / (t - Z)-Id0' (t) •

fhis implies

y'+1lb(iy) I --t

ICII for y 1 o.

For sufficiently large exponents the powers of the model operatorAtp have a simple matrix form which is independent of the numbers cj,j = 0, ... , I. If I ~ 2k, the operator A; with n ~ k + r is given by

tn.

(3.3)

0 0 (.,t n I).,. (., t n - 2).,.

., tn-A! ).,.

tn- -I

...

tn-I

...

(1, t n -A!-2).,.

0 0 (1, t n -AI).,. (1, tn-A!-I).,.

...

(l,t n 2).,. (1, t n - 3).,.

: 1, t n - 2A! ).,.

1, t n - 2A!+1 ).,.

. ...

( 1 ,tn-A!-I .,.

t n-

(l,t n

k

I).,.

0 0 0

0 0 0

0

0

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If we agree that in the case 1 ~ 2k - 1 the third row and column in (3.3) disappear, then (3.3) gives also the matrix representation of A: for n ~ k, 1 ~ 2k - 1. Let E be the spectral function of AlP and let 6. be an arbitrary interval with 0 ¢ 6.. Then making use of (3.3) and the fact that E (6.) can be written as the strong limit of a for odd n, or applying the Stieltjes-Livshic inversion formula sequence of polynomials of to the model operator of the resolvent of AlP' we obtain the following matrix representation for E(6.):

A:

t- k x~

x~·

0 0

t- k+l x~ 0 0

(-,X~t 1)00

(I,X~t-k-l)oo

(I,X~t-k)oo

(-, X~ t- 2)oo

(1, X~t-k-2)oo

(1 ,X~ t- k- 1 ) 00

(-,X~t-k)oo

(l,x~t-2k)oo

(1, X~ t- 2k +1)oo

...

t-

x~

0 0 0

0 0 0

0

0

(l,x~t-2)oo (I,X~t-3)oo

...

(1, X~t-k-l)oo

Again, if 1 ~ 2k - 1, the third row and column disappear. It follows that

IIE(6.)11 = o(Jr 2k du(t», I:;.

if a boundary point of 6. approaches zero. The growth of liE (6.) II is determined by k and independent of 1. In particular, the point Q = 0 is a regular critical point of AlP ([10], [14]) if and only if k = o. Evidently, by an appropriate choice of u, k, 1 and Cj,j = 0, ... ,1, examples for selfadjoint operators in Pontrjagin spaces with different growth properties for the resolvent and the spectral function can be constructed. 3.3. Cyclic selfadjoint operators in Pontrjagin spaces. Let A be a bounded cyclic selfadjoint operator in the Pontrjagin space n. Recall that A is called cyclic, if there exists a generating element u E n such that

n = c.l.s.{Aju : j

= 0,1, ... }.

If 4> E :F, then, evidently, the operator A", is cyclic in n (4)) with generating element 1 (or, more exactly, the element ofn (4)) corresponding to 1). The following theorem, which is the main result of this subsection, states that in this way we obtain all bounded cyclic selfadjoint operators in Pontrjagin spaces. In the following, "unitary" and "isometric" are always understood with respect to Pontrjagin space inner products. THEOREM 3.4. Let A be a bounded cyclic selfadjoint operator in a Pontrjagin space (n, [.,.J) with generating element u. Then the linear functional

4>: 'P 3 P -

[P(A)u,uJ

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273

belongs to F and A is unitarily equivalent to the operator AI{> of multiplication by the independent variable in n (q,). PROOF. If P E 'P, we have

(3.4)

[P(A)u,ul = [P(A)E (0" (A)

n IR)u,ul + [P(A)E (0" (A) \IR)u,ul

where, for a spectral set 0" of A, E (0") denotes the corresponding Riesz-Dunfor~ projection. Further,

(3.5)

[P(A)E(O"(A)\IR)u,ul =

E

([P(A)E({.8})u,ul

+ [P(A)E({P})u,u])

IIEq;(A) n(J+

and for P E 0' (A) \IR there exists a vII E IN such that (A - PI) "/I E ({P}) = O. Hence "/1-1

(3.6)

E v!-l p(")(P)[(A - PI)" E ({P}) u, ul (p E 'P).

[p (A) E ({P}) u, ul =

,,=0

Moreover, for v = 0, ... , vfJ - 1 we have

[(A - PI)" E {P}) u, ul = [(A - PI)" E ({P}) u, ul.

(3.7)

From (3.5), (3.6) and (3.7) it follows that the functional p -+ [P(A) E

(0" (A) \IR) u, ul

~elongs to F«(J\IR). Therefore, by (3.4), in order to prove that ,how that the functional

'P : 'P :3 P -+ [p (A) Uo, uol,

q, E F it is sufficient to

Uo:= E (0" (A) n IR) u,

belongs to F(IR). Denote by Po a definitizing polynomial of Ao := Alno, no := E (0" (A) n IR) n lirith only real zeros, say all ... ,a .. (mutually different) of orders 1-'11 ••• ,1-' .. , which is .~onnegative on IR (see, e.g., [14]). Let n

(Po (t) )-1 =

Pi

E E Cij (t -

ai)-j,

t E IR\{al, ... ,a.. }.

i=1 j=1

Then, for arbitrary p E 'P,

{8.8)

p(t) = 9(tiP) + Po (t)h(tiP),

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Jonas et al.

where n

~i

9(tiP) :=Po(t)LLcii(t-ai)-i(p(ai) i=l i=l n

+ ... + (j-l)!-lp(i-l)(ai)(t- a i)i- l ),

I'i

h (tiP) := L L cii (t - ai)-ip{a,.i}(t). i=l i=l We choose a bounded open interval 6. which contains u(Ao), denote by I-' the maximum of the I-'i, i = 1,2, ... ,n, and consider the set S of all polynomials P such that

°

SUp{lp(k)(t) I : 5 k 5 1-', t E 6.} 5 1. The polynomial 9 (·iP) depends on P only through the numbers p(i)(ai),i = 1, ... ,n, j = 0, ... , I-'i - 1, therefore it holds sup{l[g(AoiP)uo,uoJI : pES}
where we always assume that these data have all the properties mentioned in Lemma 1.2.

276

Jonas et al.

As k (k) is the length ofthe isotropic part ofthe Jordan chain of AlP (A"o) at 0, we have k = k. Further, if 1 ~ 2k, then 1- k + 1 is the maximal length of a Jordan chain of AlP at 0, hence in this case 1 = It follows that the sizes of the blocks of the matrices for AlP and A"o in (3.1) or (3.2) and for the Gram operators G IP and G"o from (2.6) or (2.7) coincide:

t.

t· (4.2)

AlP =

[

~

E Sl J

E' D Here we agree that in case r = 0 the third row and column disappear. The blocks in (4.2) can be read off from (2.6), (2.7), (3.1), (3.2) with a = OJ we only mention that D = 0 if r > O. The corresponding blocks of A"o are denoted by iI1, Z etc. The operator U in (4.1) is partitioned in the same way: U = (Uij)t. As U maps the algebraic eigenspace of AlP at 0 onto that of A"o and these subspaces are given by the vectors with vanishing first and second block components, it follows that Un = U14 = U2S = U24 = o. Also the set of vectors with vanishing first, second and third block components is invariant under AlP and A"o, hence US 4 = o. Now we write the relation (4.1) for the block matrices of AlP' A"o, U. Considering the components ·21 on both sides it follows that SlU21 = U21 t·, hence U21 = 0 and, similarly, U3l = o. Then (4.1) turns out to be equivalent to the following relations:

(4.4)

i· Ull = Ull t·, i· Ul2 + EU22 =

(4.5)

SlU22 = U22 Sl,

(4.6)

JU22

(4.7) (4.8)

S2U3S = USS S 2, A, A E Ull + SlU41 = U41 t· +U44 E',

(4.9)

E'Ul2

(4.3)

(4.10)

(4.11)

+ S2US2 =

UllE + Ul2 Sl, US2 S1 + UssJ,

+ DU22 + GUS2 + SlU42

= U41 E

+ U42 S1 + U4S J + U44 D,

GUss + SlU43 = U43 S 2 + U44 C, SlU44 = U44 S 1.

As U is isometric with respect to the inner products on L2 «(1') EB (l21t+r and L2 (u) EB C 21t+ r , generated by the Gram operators G IP and G"o, respectively, we have also

(4.12)

U*G"oU = G IP ,

which is equivalent to

(4.13)

Ui1UU = I,

2'n

Jonas et al.

(4.17)

+ U41 Z U22 = 0, Ui2U12 + U22(HIU22 + H3U32 + Z U42 ) + U;2(H;U22 + H2U32 ) + U42 Z U22 = U22 (H3U33 + Z U43 ) + U32H2U33 = H 3, U22 Z Uu = Z,

(4.18)

U33H2U33 = H 2.

(4.14) (4.15) (4.16)

Uil Ul2

Ht,

The relations (4.3) and (4.13) imply that the operators of multiplication by the independent variable in L2 (17) and L2 (17) are unitarily equivalent. As is well-known this means the equivalence of the measures 17 and 17 and that U11 is the operator of multiplication by a u-measurable function i such that lil 2 = dl7/du ([1]). Next we consider (4.5). It implies that U22 is a Toeplitz matrix of the form

U22 =

0 Uo

0

Ul

UIc-l

UIc-3 UIc-2

Uo Ul

[~1c:-2"

0

!j.U;Em,

j = O, ... ,k-1.

Uo

Further, writing U12 = (VlcVIc-l ..• vI) with Vj E L 2 (u),j = 1,2, ... ,k, the relation (4.4) is equivalent to

A = t-1 (A'"Y - Uo A) , V2 A = t- 1 (AVI - Ul A) , ... , Vic A = t- 1 (AVIc-l - UIc-l A). VI

(4.19) This implies

(4.20) u-a.e. That is, if k > 0 the function i(E L 2 (u» has a well-defined "value" Uo at t = 0 and also "derivatives" up to the order k -1. That uo, Ul, ... , 'LIc-l are uniquely determined by the relations (4.19) or by (4.20) follows from the condition Jr 2 du(t) = 00. Similary, making use of( 4.11) and (4.8) and putting U41 =

«., vdu, . .. , (., VIc)u »T

and

[ UlU,

0 Uo

Uo

!1 E~,

Ul

Uo

0 0

U44 = UIc:-2 UIc-l

ve find

,u;

UIc-3 UIc-2

j=O, ... ,k-l,

Jonaset at.

278

and all these functions belong to L2(0"). It is easy to see that the relation (4.17) is now automatically satisfied. In particular, we have Uo f:. 0 and •

-1

'Ito = 'Ito •

Then by (4.20) there exists a polynomial p of order :S k - 1 with real coefficients and p(O) = luol such that t-Al(lil- p) E L2(U). In particular, we have (4.21) We summarize some of the above results to a necessary condition for unitary equivalence of A", and A.,o. Here It (and correspondingly k) are given by (2.8). THEOREM 4.1. Let cp, I{; E F(IR; 0) with representation according to Lemma 1.2. If the operators A", and A.,o are unitarily equivalent, the following statements hold: (i)

It

= k, k = k,

and, if I ~ 2k, 1 =

i.

(ii) The measures 0" and u are equivalent; if k > 0 there exists a polynomial p of order :S k - 1 with p(O) f:. 0 and real coefficients such that with fJ := (du/dU)l the function belongs to L2(U). The meaning of the necessary conditions (i), (ii) in Theorem 4.1 is enlightened by the following result, which, in fact, contains Theorem 4.1. THEOREM 4.2. Let CP,I{; be as in Theorem 4.1 and denote by E,E the spectral functions of A"" A.,o, respectively. Then the conditions (i), (ii) in Theorem 4.1 are necessary and sufficient for the unitary equivalence of the spectral functions E and E. PROOF. 1. Assume that E and E are unitarily equivalent. Then, evidently, It = k. The number k (k) coincides with dimension of the isotropic part of the closed linear span £(0) (£(0» of all ranges of E(A)(E(A», where A is an arbitrary interval with 0 ¢ A. ~ .~ Therefore k = k. The orthogonal companion £(0) (£(0» of £(0) (£(0» is the algebraic eigenspace of A", (A.,o) at o. Then, in view of the results of Section 3.2, we have 1 ~ 2k if and only if i ~ 2k, and in this case 1 = i. Hence the condition (i) holds.

.

.

In order to prove (ii) set n := 2( k + r) + 1 with r = max{O, 1- 2k + 1}. Then A~-l is nonnegative with respect to the Pontrjagin space inner product, and A~ = ftndE(t) (d., e.g., [7; 3.3]). A similar relation holds for A;. Therefore the operators A~ and A~ are unitarily equivalent. Assume that Ie > (if Ie = 0, a similar, much simpler reasoning applies). We write A~ in a form similar to that of A", in (4.2) (see (3.3»: t· is replaced by tn., E = (t..- AI ••• t .. - l ), 8 1 = J = 8 2 = C = 0, E' = «.,t..- l ) ..... (.,t.. -·).. )T and

°

Jonaset aI.

279

(1, t,,-l )C7] (1, t,,-2)C7

(1,

(1, t,,-21o+ 1)C7

t,,~le-1)C7

.

Let U = (Uij)t be a unitary operator from (L2(0') EB(:21o+",(G'P"')La(C7)E!)(!21o+ o. With pet) = Uo " ..VTe -

+ Utt + ... + UTe_tt Te - t

t- Te ("9 - P )" - t- i ("u,·ti , v,.•.-

Then the operators Un =

g., U12

we associate functions Vt, ••. ,VTe E L2(U) :

"tTe-t + t Te VTe, ") 3. -1 + ... + Uk-t - , ••• , Ie -

1•

:= (Vk •.. Vt) and

:~ U22 := [

o Uo

Uk:-t

satisfy the relations (4.13), (4.23) and (4.24). From (ii) it follows that there exists a (uniquely determined) polynomial q(t) = Uo + utt + ... + UTe_tt k - t with real coefficients and Uo '" 0 such that the function

belongs to L2(U). It is easy to see that the coefficients of p and q satisfy the relations

i

(4.28)

uouo = 1,

L uiui-i = 0,

j = 1, ... , Ie - 1.

i=O

Define functions

VI, •• • , VTe-t

E L2 (u) by

Then with the operators U41 := «.,vt} ...... (.,VTe) ... )T,

U44

:=

[

:~

o Uo

u/r:-t the relations (4.28) are equivalent to (4.17) and (4.14), (4.25) and (4.26) are satisfied. In order to prove Theorem 4.2 for Ie > 0 it remains to find operators Un, Usa, U42 and U4S which fulfil the relations (4.15), (4.16) and (4.18). Since by (i) the nondegenerate

Jonas et aI.

281

forms (H2 ·, .) and (il2 ·, .) have the same signatures, there exists a matrix U33 which satisfies (4.18). We set U32 = O. Consider the equation (4.15):

Evidently, the operator U42 := HU;2Z)-1 S satisfies this relation. If we choose

then the equation (4.16) is fulfilled.

= 0, then the operator

If k

U= [Uno U0] 33

with Un and U33 as defined above has the required properties. This completes the proof. REMARK 4.3. It is easy to see that for k = k = 0 the conditions (i) and (ii) in Theorem 4.1 are also sufficient for the unitary equivalence of AI" and A",. Necessary and sufficient conditions for the unitary equivalence of AI" and A", can be given also under other additional assumptions. However, a complete treatment of the relations (4.3) - (4.11) and (4.13) - (4.18) seems to be complicated. In the following subsection we consider the case /(, = 1. 4.2. The case /(, = 1. Let 'P, cp E F (IR; 0) be such that the numbers /(', It given by (2.8) are one. By Remark 4.3 we can restrict ourselves to the case k = k = 1. THEOREM 4.4. Let If I

'P,cp E F(IR,O),

/(, = It = 1 (see (2.8)) and k =

k=

1.

= i = 2, then the operators AI" and A", are unitarily equivalent if and only if

(i) the measures u and iT are equivalent,

(ii) with

t

(iii)

-+

g :=

(du/diT)! there exists a nonzero real number go such that the function

r1 (g (t) - go) belongs to L2 (iT),

lyol2c2 = C2. If I, i :s 1, then AI" and A", are unitarily equivalent if and only if condition (i)

and

·the following condition are satisfied. (iv) There exists a complex function i E L2 (iT) with lil 2 = du/diT and a nonzero number io such that the function t -+ t -1 (i (t) - io) belongs to L2 (iT) and

I4.29)

282

Jonas et al.

PROOF. If k = 1 and r ~ 1. then for the blocks in (4.2) we have Sl = S2 = 81 = 82 = 0 and (if r = 1) J = j = 1. Consider first the case I = 2. that is r = 1 (or C2 > 0). As above the relations (4.3). (4.4) and (4.13) imply (i) and (ii). and (4.5) - (4.11) become

(4.30)

U22 = U33 .

(4.31)

(Uu ·.l)u=Un t·+U44 (·.1) ....

(4.32)

(U12 .• 1)u + C2U32 = Unl

(4.33)

C2U33 = U44C2.

+ U43 •

Further. (4.14) - (4.18) are equivalent to

+ U;lU22 = o.

(4.34)

U;lUl2

(4.35) (4.36)

+ U22cOU22 + U;2ClU22 + U;2U22 + U22clU32 + U;2C2U32 + U22 U42 = Co. U22ClU33 + U;2C2U33 + U22 U43 = CIt

(4.37)

U22 U44

(4.38)

U;3C2U33 = C2.

U;2U12

= 1.

The relations (4.30) and (4.38) give U22U22C2 = C2. As above (see (4.21» we find U22 U22 = 1901 2 which proves (iii). Assume now that I = i = 2 and the conditions (i). (ii) and (iii) hold. Then, if Uu := 9·,Ul2 := t- l (9 - 90). U22 = U33 := 90, Un := (·,vI) ... with VI := t- l (9- 1 90 1 ). U44 := 90 1 the conditions (4.3), (4.4), (4.13) and (4.30). (4.31), (4.33), (4.34), (4.37) and (4.38) are fulfilled. There remain (4.32). (4.35) and (4.36) to be satisfied which give three equations for the three numbers U32 ,U42 and U43 • The relations (4.32) and (4.36) lead to the following system of equations for (real) U32 • U43 :

f

UndO- + C2 US2 =

U22ClU3S

f

vldu + U43 •

+ U32C2US3 + U22 U43

=

Cl·

Its determinant of the coefficients of US2 • U43 is

C2 det [ .U C

33

U- 1 ] =

22

2c290 -:j:. 0,

hence Un and Uu are uniquely determined. Finally U42 follows from (4.35). and AlP and A.,a are unitarily equivalent. If r = 0 (or C2 = 0). in the block matrices in (4.2) the third rows and columns disappear and D = CI. iJ = CI. Assume that A

of an operator space S into another is said to be unital if 4>(1) = 1 holds; contractive, ~o8itive

if

114>(z)1I :5 IIzll (z 4>(z)

~

E S),

0 (0:5 z E S)

Okayasu

288

holds, respectively; completely contractive, completely positive if the tensor product t/J ® id m of t/J and the identity map id m of the m x m matrix algebra Mm is contractive, positive, resectively, for any m

~

1.

It is fundamental that any unital contractive map of an operator system into another

is positive, that a positive map t/J of an operator system into another is bounded (in fact,

Iit/Jil

~

211t/J(1)ID, and that a positive map t/J of a C*-algebra A into a C*-algebra B is

completely positive if either A or B is abelian. The Steinspring theorem [8] asserts that any unital, completely positive map t/J of a unital C*-algebra A into the C*-algebra B(1t) on a Hilbert space 1t has a canonical representation, namely, there exist a Hilbrt space (unique up to unitary equivalence) /C ;2 1t and a *-representation 'II" of A on /C such that

'11"(04)1£ is dense in /C and

t/J(z)

= P'II"(z)lll

(z

E A),

where P is the projection onto 1t. Now we want to give a proof of Theorem 1. In it, Arveson's extension theorem [3] (See

[7]) is essential; it asserts that, any unital completely contractive map of an operator space S into an operator space T extends to a completely positive map of any C* -algebra A ;2 S into a C*-algebra B ;2 T.

Proof of Theorem 1. Assume that T 1 , •.. ,Tn are commuting contractions on a Hilbert space 1t and satisfy the von Neumann iequality in the strong sense. By assumption the linear map

of the operator space p(Tn) of all polynomials in n variables eiBl , ••• ,eiB ", on Tn T x ...

X

=

T, into B(ll), is unital and completely contractive. Then it extends to a unital,

completely positive map of the C*-algebra C(Tn) of complex-valued continuous functions on Tn, into B(ll). Therefore, there exist a Hilbert space /C ;2 1t and a *-representation 'II" of C(Tn) on /C such that

t/J(f) = P'II"(f)lll (f E C(T n

P the projection onto 1t. Put Uk

= 'II"(eiB1 )

(k

»,

= 1,··· ,n).

Then Ut.··· ,Un are unitary

operators and satisfy that

Tf'l ... r;:'''

= PUf'l ... U:''' 11t (ml,···, mn ~ 0).

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289

Conversely, let /C be a Hilbert space ;2 1i and U1 , ••• , Un be commuting unitary operators on /C such that

P the projection onto 1i. Consider the *-homomorphism ~ of the *-algebra of all polynom ials in variables ei'l , e- i91 , ... , ei'n , e- i9n , on Tn , to B('IJ) '" such that

for k = 1, ... , n. We can see that it is bounded and satisfies the inequality

for any p. Therefore, by the Stone-Weierstrass argument, it extends to a *-representation of C(Tn). So,

~

is completely contractive. Consequently, we have II(Pij(Tt,·.· , Tn))! 1 ~ II (Pij (U1 , ••• , Un»11 = II(~ ® idm)((Pij))!1 ~ 11(Pij )11

=

for any m x m matrix (Pij), Pij polynomials in variables ei91 , ... , ei9n , which completes the proof. 3. An effect of generation of nuclear algebras Next, we will give a Proof of Theorem 2. It is sufficient to find a unital completely contractive map of

p(Tm+n) into B(1i), which maps each variable ei9j , ei9k to Sj, TAl> respectively. We already have, via Theorem 1, unital completely contractive maps

~l

of p(Tm) into

B(1£) so that ~l(ei'j) = Sj, ~2 of p(Tn) into B(1£) so that ~2(ei'k) = Tk. According to Arveson's extension theorem, ~l (resp. ~2) extends to a unital completely positive map tPl (resp. tP2) of C(Tm) (resp. C(Tn» into.A (resp. 8), where.A (resp. 8) is the C*-algebra

Okayasu

290

generated by S1,'" ,Sm (resp. T1,'" ,Tn). It can be seen that the tensor product tP1 ®tP2 of tP1 and tP2 is a unital, completely positive, and so completely contractive, map, of the C*-tensor product C(Tm) ® C(Tn) of C(Tm) and C(Tn) (which can be thought of as C(Tm+n)) into the minimal C*-tensor product A ® E of A and E, i.e., the completion of A 0 E under the operator norm II II considered on A 0 E (which is known to be the smallest among all C*-norms on A 0 E [10]). Hence, the tensor product 4>1 ® 4>2 of 4>1 and 4>2 , the restriction of tP1 ® tP2 to p(Tm+n) (identified with p(Tm) 0 P(T n )), is a unital completely contractive map of p(Tm+n) into A 0 E. Consider then the *-representation 4> of A 0 E on 'It such that

4>(X ® Y) = XY (X E A, Y E E). Since the operator norm II lion A0E coincides, by assumption, with the (largest) C*-norm

II

II" defined by the identity IIVII" = sup{II1I'(V)11 : 11' is a *-representation of A 0 E},

for each V E A 0 E (See [7]), we have the inequality IIL:X',Ykll ~

k

IIL:Xk®Ykll" k

=

IIL:Xk ®Ykll, k

for X k E A and Yk E E. This shows that 4> may extend to a *-representation of the C* -algebra A ® E. Hence, as above, 4> is completely contractive. It is obvious, on the other hand, that 4> is unital, so, the composition 4> 0 (4)1 ® 4>2) of 4> and 4>1 ® 4>2 is unital and completely contractive; and maps each variable ei8j , ei8l to Sj, Tk, respectively. Now the proof is complete.

References 1. T. Ando, On a pair of commuting contractions, Acta Sci. Math. 24(1963),88-90.

2. T. Ando, Unitary dilation for a triple of commuting contractions, Bull. Acad. Polonaise Math. 24(1976), 851-853. 3. W. B. Arveson, Subalgebras of C*-algebras, Acta Math. 12;1(1969), 141-224.

Okayasu

291

4. M. J. Crabb and A. M. Davie, von Neumann's inequality for Hilbert space operators, Bull. London Math. Soc. 7(1975), 49-50 5. T. Okayasu, On GCR-operators, Tohoku Math. Journ. 21 (1969), 573-579. 6. S. K. Parrott, Unitary dilations for commuting contractions, Pacific Journ. Math. 34(1970), 481-490. 7. I. Paulsen, Completely bounded maps and dilations, Pitman Res. Notes Math. Ser. 146,1986. 8. W. F. Steinspring, Positive functions on C·-algebras, Proc.

Amer.

Math.

Soc.

6(1955), 211-216. 9. B. Sz.-Nagy and C. Foi/l.§, Harmonic analysis of operators on Hilbert space, Amusterdam-Budapest, North-Holland, 1970. 10. M. Takesaki, On the cross-norm of the direct product of C*-algebras, Tohoku Math. Journ. 16(1964), 111-122. 11. N. Th. Varopoulos, On an inequality of von Neumann and an application of the metric theory of tensor products to operator theory, Journ. Funct. Analy. 16(1974),

83-100.

Department of Mathematics Faculty of Science Yamagata University Yamagata 990, JAPAN

MSC 1991: Primary 47A20, 47A30; Secondary 46M05

292

Operator Theory: Advances and Applications, Vol. 59 © 1992 Birkhiiuser Verlag Basel

INTERPOLATION PROBLEMS, INVERSE SPECTRAL PROBLEMS AND NONLINEAR EQUATIONS

L. A. Sakhnovich

The method of operator identities of a type of commutation relations is shown to be useful in the investigation of interpolation problems, inverse spectral problems and nonlinear integrable equations.

Suppose that the operators A, S, 'PI, 'P2 are connected by the relation (1)

where G 1 and H are Hilbert spaces, dim G 1

< 00,

and {HI, H 2 } is the set of bounded operators acting from HI to H 2 • We also introduce the operator J E {G, G} where

J=[OE1

E1 ]

o .

Formula (1) is a special case of the operator identity of the form (2)

which is a generalization of the commutation relations and it also generalizes the wellknown notion of the node (M. S. Livsic [1] and then M. S. Brodskii [2]). The identities of the form (2) proved to be useful in a number of problems (system theory [3], factorization problems [3], interpolation theory [4], the method of constructing the inverse operator T = S-l [5], the inverse spectral problem [3] and theory of nonlinear integrable equations [6]). There are close ties between all these problems and corresponding results.

293

Sakhnovich

In the present paper we shall consider three of these problems: interpolation problems, inverse spectral problems and nonlinear integrable equations. 1. Let £ be a collection of monotonically increasing operator-functions and

r( u) E {G 1 , Gd is such that integrals (3) (4)

converge in the weak sense. Then the integral

(5) also converges in the weak sense. Let us introduce the operators (6)

where

and operator F from {G 1 , H} is defined by the equality (7)

Now we shall formulate the interpolation problem which is generated by operation identity

(1) [4]. It is necessary to describe the set of r( u) E £ and a = a*, ;8iven operators S, 1'1 admit the representation

S =

S,

1'1 = 1'.

f3

~

0 such that the

(8)

Let us note that according to (3) the necessary condition of the formulated problem is the inequality

S ~s

~

o.

(9)

an example we shall consider the bounded operator S in the space L2(0, w) ofthe form (SI)(z)

r

d = dz 10

I(t)s(z - t) dt.

(10)

Sakhnovich

294

Then the equality ((AS - SA*)/)(z)

=i

1'"

/(t)[M(z)

+ N(t)) dt

(11)

is valid. In equality (11) M(z)= s(z),

N(z)=-s(-z),

O:5z:5w

and the operators A, A* are defined by the equalities (A/)(z)

=i

1"

/(t) dt,

(A* /)(z)

= -i

1'"

/(t) dt.

Formula (11) is a special case of the operator identity of form (1), where

and 9 are constants. The corresponding interpolation problem has the form. It is necessary to describe the set of r(u) E £ which gives the representation

(S/,/) =

I: 11'"

/(z)e- iu " dzl2 dr(u).

If the operator S has the form (S/)(z) = [

/(t)lI:(z - t) dt

we come to the well-known Krein problem: it is necessary to describe the set of r( u) which gives the representation lI:(z) =

i:

ei"u dr(u).

In our approach to the interpolation problem we use the operator identity and operator form of Potapov inequality [4). Operator identity (11) gives a tool for constructing the inverse operator T = S-I. The operator T can be found in the exact form by means of the functions N l (z), N 2 (z) which are defined by the relations

We have proved that the knowledge of N l , N2 is that minimal information which is necessary for constructing T [5]. 2. Let us consider the inverse spectral problem which is connected with operator identity (1).

295

Sakhnovich

THEOREM. Suppose that the following conditions hold: 1. S is positive and invertible.

II. There exists a continuous increasing family of orthogonal projections PC, 0 ::;; (::;; w, Po

= 0,

Pw

=E

such that A* P, = P,A* PC.

III. The spectrum of A is concentrated at zero. Then the following representations hold .r'o

w«(, z) =

la' exp[izJ dUl(t)l

where w«(, z) = E II

+ izJ II* S,l(E - zAc)-l P,II,

= [PI, P 2],

If Ul (x) is absolutely continuous, then

S,

= P,SP"

u~ (x) ~

A,

= P,AP,.

0 and

~: = izJu~(x)w(x, z),

w(O, z)

= E.

(12)

':Janonical system (12) corresponds to operator identity (1). We shall introduce the main definitions. Let us denote by the space of vectorfunctions with the inner product

We define the function

and the operator

Vg =

/2.

monotonically increasing m x m matrix-function T( u), -00 < U < 00 will be called a Ipectral function of system (12) ifthe operator V maps L2(Ut} isometrically into L2(T). ~

296

Sakhnovich

THEOREM. The set of r( u) which are solutions of the interpolation problem and the set of spectral functions of the canonical system coincide. The following results give a method for solving the inverse spectral problem

[3]. Let us suppose that the operators A and P2 are fixed. In this way we define the class of canonical systems (12). Then let us suppose that the spectral data of system (12) are given, i.e. the spectral function r( u) and the matrix interpolation formulas we have:

PI

= -i roo

S=

Loo [A(E -

I:

uA)-l

Q

are known. Using the

+ ~E]P2 dr(u) + iP2

Q

l+u

(E - uA)-lP2[dr(u)]P;(E - uA*)-l

II = [PI, P 2 ],

O'l«) = 11* S,l Pell.

(13) (14)

(15)

These formulas (13)-(15) give the solution of the inverse spectral problem.

Ie:

If (A/)(z) = i /(t) dt, P29 = 9 we come to the well-known inverse problems for the system of Dirac type. In the general case when (A/)(z) = iW

L"

W = diag{wl,'"

/(t) dt ,Wn }

we come to the new non-classical inverse problem. The necessity of investigating nonclassical problems is dictated both by mathematical and applied questions (interpolation theory, the theory of solitons). Under certain assumptions formulas (13)-(15) give the solution of the inverse spectral problem in the exact form [3]. Let us introduce an analogue of the Weyl-Titchmarsh function v(z) for system (12) with the help of the inequality

The matrix function v( z) belongs to the Nevanlinna class, i.e.

v(z) -. v*(z) ~ 0,

Imz >

o.

I

The connection of v(z) with the spectral data r(u) and

v(z) =

Q

+

Q

is the following [3]

L: C, ~ 1:u2) z -

dr(u).

Sakhnovich

297

We have considered the case when 0 5 x < 00. As in the case of Sturm-Liouville equation, the spectral problems on the line

(-00 < x < 00) can be reduced to the problems on the half-line (0,00) by doubling the dimension of the system. The problem on the line contains the periodical case. 3. The method of inverse scattering problems is effectively used for investigating the nonlinear equations (Gardner, Kruskal, Zabuski, Lax, Zaharov, Shabat [8]). The main idea comes to the following. The nonlinear equation is considered together with the corresponding linear system. The evolution of scattering data of the linear system is very simple. Then by using the method of inverse problem the solution of nonlinear system can be found. The transition from the inverse scattering problem to the inverse spectral probem removes the demand for the regularity of the solution at the infinity and permits to :onstruct new classes of exact solutions for a number of nonlinear equations [7]: Rt =

i(R..,.., - 21RI2 R)

Rt =

-41 R..,..,,,, + "23 1R 12 R..,

(NS)

(16)

(MKdV)

(17)

(Sh-G)

(18)

{Pcp {)x{)t = 4 sh cpo

fhese equations have found wide applications in a number of problems of mathematical "hysics. The corresponding linear system has the form

~: = izH(x, t)w,

w(O, t, z)

= En.

(19)

~ the case of Sh-Gordon equation we have [8]

H x t _ [ ( , )-

0 exp[cp(O, t) - cp(x, t)]

exp[cp(x, t) - cp(O, t)]] 0 .

tet vo(z) = v(O, z) of corresponding system (19), (20) be a rational function of z: i.e. N

vo(z) = i -

L f31:,o/(z + ;0'1:,0). 1:=1

'Then v(t, z) is also a rational function N

v(t, z) = i -

L f31:(t)/[z + iO'I:(t»). 1:=1

(20)

Sakhnovich

298 Let us write down v(t, z) in the form

where N

N

Pt(t,z) = II[z-ak(t)],

P2(t, z) = II[z - Vk(t)).

k=l

k=l

Let us introduce

It is essential that unlike Pl(t, z), P2(t, z) the coefficients Q(z) do not depend on t. It

means that the zeros of the Q(z) do not depend on t either. Let the inequality Wj =f:. Wk be true when j =f:. k. Let us number the zeros of Q(z) in such a way that Rewj > 0 when 1 ~ j ~ N. The solution of the Sh-Gordon equation which corresponds to the rational vo(z) is as follows

where

This result was obtained jointly with post-graduate Tidnjuk [7). The corresponding solution R( x, t) of equations (16), (17) has the following form

R(x,t) = -2(-I)N ~dx,t)/~2(X,t), 1

1

1

N-2 w 2N ;2N

W2N;2N

Sakhnovich

299 1

1

1

N-l

W 2N

12N

where II:

=

CI:

exp2(wl:z - Ol:t),

01: = -iw~

(NS),

01: = w~

(MKdV).

4. If Wj = wj, O!j,O = -O!j,O then the corresponding solution R(z, t) of MKdV is real. All the real singularities of R(z, t) are poles of the first order with the residues +1 or -1. When t

-+

±oo,

the solution R(z, t) is presented by a sum of simple waves N

R(z,t) ~ LRt(z,t),

t

-+

±oo

(21)

j=l

Rt(z, t) = 2(-1)jwj/sh[2(wjz -

w7 t + ct»)·

(22)

The considered nonlinear equations do not have N -soliton solutions. The constructed solutions are similar to the N -soliton solutions. The behaviour of the singularities of the solutions is analogous to the behaviour of the humps of the N-soliton solutions and can be interpreted in the terms of a particles system. We have proved that the corresponding "articles system is a completely integrable one with the Hamiltonian (23)

where Pj Pj

= wJ (MKdV), = 2Imwj (NS)

qj = Pjt

+ Cj.

1

Pj = - 2

(Sh-G)

(24)

Wj

(25)

(26)

The variables Pj, qj are variables of the action-angle type. It follows from formulas (21), (22) for MKdV that Pj coincides with the limit velocity of the wave. The same situation ,is in Sh-G and NS cases.

Sakhnovich

300

MKdV Wl

= 0.3

al = 0.35

W2 a2

= 0.5 = 0.45

W3

a3

c=k>O Fig. 1

= 0.7 = 0.55

301

Sakhnovich

Sh-G Wt

= 1

at = 1.2

W3

=6

a3 = 2.2

c=k>O Fig. 2

Sakhnovich 302

Sh-G Wl Ql

= 1 0.8

=

W3 Q3

= 6 = 5.S

Fig. 3

Sakhnovich

303

It also follows from formulas (21), (22) that the lines of singularities have N

asymptotes (MKdV) WjZ -

wft + cT = 0,

1:5 j :5 N,

Let us number these asymptotes when t

--+ -00

the same lines of singularities when t

+00.

--+

t

--+

±oo.

by the order of velocity values and consider

Then the corresponding asymptotes are again

ordered by the velocity values but in the opposite direction (Fig. 1). It means that the particles exchange their numbers. In the case of Sh-Gordon equation the situation is the same (Fig. 2). The particles exchange their numbers even if there is no crossing (Fig. 3). The particles can be of two kinds: plus particles if the corresponding residues are +1, and minus particles if they are -1. The particles of the same kind don't cross. In Fig. 3 the particles are of the same kind. 5. The considered equations generated self-adjoint spectral problems. If we

study the equations i

Rt = '2(R",,,, Rt =

2

+ 21RI R)

13

-:t R",,,,,,, - '2IRI 2 R",

8 2 1{' 8z8t = 4 sin I{'

(NS)

(27)

(MKdV)

(28)

(sin-G)

(29)

then the nonself-adjoint spectral problems correspond to them. The analogue of the WeylTitchmarsh function for this case was introduced and the analysis of the equations of the form (27)-(29) was done by A. L. Sakhnovich [9], [10].

REFERENCES 1. M. S. Livsic, Operators, oscillations, waves (open systems), Amer. Math. Soc., 1966. 2. M. S. Brodskii, Triangular and Jordan representations of linear operators, Amer. Math. Soc., 1971. 3. L. A. Sakhnovich, Factorization problems and operator identities, Russian Math. Surveys, 41 no. 1 (1986), 1-64. 4. T. S. Ivanchenko, L. A. Sakhnovich, An operator approach. to th.e investigation of interpolation problems, Dep. at Ukr. NIINTI, N 701 (1985), 1-63.

Sakbnovich

304

5. L. A. Sakhnovich, Equations with a difference kernel on a finite interval, Russian Math. Surveys 35 no. 4 (1980), 81-152. 6. L. A. Sakhnovich, Nonlinear equations and inverse problems on the semi-axis, Preprint, Institute of Math., 1987. 7. L. A. Sakhnovich, I. F. Tidnjuk, The effective solution of the Sh-Gordon equation, Dokl. Akad. Nauk. Ukr. SSR Ser. A, 1990 no. 9, 20-25. 8. R. K. Bullough, P. I. Caudrey (Eds), Solitons, New York, 1980. 9. A. L. Sakhnovich, The Goursat problem for the sine-Gordon equation, Dokl. Akad. Nauk. Ukr. SSR Ser. A 1989, no. 12, 14-17. 10. A. L. Sakhnovich, A nonlinear Schrodinger equation on the semi-axis and a related inverse problem, Ukrain. Math. J. 42 (1990), 316-323. Sakhnovich, L. A. Odessa Electrical Engineering Institute of Communications Odessa, Ukraine MSC 1991: 47 A62, 35Q53

305

Operator Theory: Advances and Applications, Vol. 59 © 1992 Birkhiiuser Verlag Basel

Extended Interpolation Problem in Finitely Connected Domains SECHIKO TAKAHASHI

This paper concerns the matrix condition necessary and sufficient for the existence of a function I, holomorphic in a finitely connected domain and having III ::; 1 and finitely many first prescribed Taylor coefficients at a finite number of given points. In a simply connected domain, some transformation formulas and their applications are given. The results of Abrahamse on the Pick interpolation problem are generalized to the above extended interpolation problem.

Introduction Let D be a bounded domain in the complex plane C, whose boundary consists of a finite number of mutually disjoint analytic simple closed curves, and let B be the set of functions f holomorphic in D and satisfying ifi $ 1 in D. In this paper we consider the following extended interpolation problem: Let Zl, Z2, ... ,Zk be k distinct points in D and, for each point Zj, let CiO,' .. , Cjni-l be ni complex numbers. For these given data, find a function fEB which satisfies the conditions nj-l

(EI)

fez)

=

L

Ci"'(Z - Zi)'"

+ O((z -

Zj)ni)

(i=I, .. ·,k).

",=0

In the Part I, we introduce, as powerful tools for the studies of this problem (EI), the Schur's triangular matrix Do used by Schur in [13} and our rectangular matrix M, which made it possible to unify Schur's coefficient theorem and Pick's interpolation theorem (Takahashi (14)). We give important transformation formulas which express the changes of these matrices under holomorphic transformations in terms of the transformation matrices. In the Part II, we recall our main results obtained in [14} and [15} for the problem (EI) in the case where D is the open unit disc. As an application of these results and the transformation formulas, we give a criterion matrix of the extended interpolation problem in the case where D is a simply connected domain in the Riemann sphere having at least two boundary points and the range W is a closed disc in the Riemann sphere. When W contains the point at infinity, we have of course to modify the conditions (EI) appropriately and the solutions may have poles.

Takahashi

306

In the Part III, we show that the results of Abrahamse in [lIon the interpolation problem in finitely connected domains can be extended to our extended interpolation problem. PART I. MATRICES AND TRANSFORMATION FORMULAS

§1. Matricial Representation of Taylor Coefficients. ~.

(1) Schur's Triangular Matrix

To a function

00

J(z) =

L ca(z -

zo)a

a=O

holomorphic at Zo and to a positive integer n E N, we assign a triangular n x n matrix ~(f;zo;n)

Co

=

Cl

J

Let g( z) be another function holomorphic at Zo, we see immediately ~(f

+ g; Zo; n) =

~(fg;

Zo; n) =

~(f;

Zo; n) + ~(g; Zo; n),

~(f;

Zo; n)·

~(g;

Zo; n)

= ~(g; Zo; n)· ~(f; Zo; n), ~(l;

Zo; n) = In

(the unit matrix of order n).

(2) Rectangular Coefficient Matrix M. To a function 00

F(z, () =

L

aaP(z - zo)a«( -

(0/

a,p=O

holomorphic w.r.t. (z, () at (zo, (0) and to (m, n) E N x N, we associate an m x n matrix

M(F;z,,~;m,n) ~ [a~:;; ..... ~:=~J

For another function G(z, () holomorphic w.r.t. (z, () at (zo, (0), we have

M(F + G; Zo, (0; m, n) = M(F; Zo, (0; m, n) + M(G; Zo, (0; m, n). Moreover, for functions J(z) and g«(), holomorphic at Zo and (0 respectively, we have the useful product formula (PF)

M(fFg; zo,(o;m,n) = ~(f; zo;m)· M(F;zo,(o;m,n)· ~(g;(o;n)*,

where by ~ * = t ~ we mean the transposed of the complex conjugate of~. This product formula can be established by a direct calculation.

307

Thkahashi

§2. Transformation Matrix and Transformation Formulas The transformation formula for the matrix M which we established in [14] is the pivot of our present studies. For a transformation z = cp( x) holomorphic at Xo with zo = cp( xo) and for mEN, we define the transformation matrix n( cPj Xo j m) as follows: Write cp(X) = zo

+ (x -

~m = ~(CPljXoj

XO)CPl(X),

m),

E~) = M«z - zo)a"«(=-----:"(o't'jzo,(ojm,m)

(a = 0,··· ,m -1)

and put m-l

n(lll· 'L...J " ~am E(a) T' Xo·, m) m . a=O

The matrix E~) is the m x m matrix whose (a + 1,0' + I)-entry is 1 and the all other entries are o. ~?. = 1m and ~::. = ~::.-l ~m (a = 1,2,·· .). The matrix n is of the form

.

1

o n(cpjXOjm) =

c

* o *

c2

*

cm -

1

where c = CPl(XO) = cp'(xo). If c :j: 0, then n(cpj Xoj m) is an invertible matrix. If cp(x) is the identical transformation, then CPJ(x) = 1, ~m = 1m and hence n(cpjxojm) = 1m. In terms of this transformation matrix n, we showed in [14] THEOREM 1 (TRANSFORMATION FORMULA FOR M). LetF(z,() be afunction holomorphic w.r.t. (z,() at (zo,(o). Let z = cp(x),( = tfJ(O be functions holomorphic at xo,{o, with Zo = cp(xo),(o = tfJ({o) respectively. Put

Then, for (m,n) E N x N, we have M(Gj xo,{ojm,n) = n(cpjxojm). M(Fj Zo, (oj m,n)· n(tfJj{Oj n)*.

Thkahashi

308

As an application of the preceding transformation formula, we obtain THEOREM

2

(TRANSFORMATION FORMULA FOR

6). Let

00

J(z) =

L co(z -

zo)o

0=0

be a function holomorphic at Zo and let

0). Let Zl,Z2,··· ,Z,. be k distinct points in D. For each Zj, let mj be a nonnegative integer and let CjO,··· , Cjn;-l be nj complex numbers (nj ~ 1). Assume CjO '" 0 if mj > o. The problem in this section is to find a meromorphic function f in D with values in W, which satisfies the conditions 1

(EI)

f(z)

n;-l

= (z-z. ). (~Cjo(Z-Zj)o+O«Z-Zj)n;») m. ~ I

(i=l,···,k),

0=0

where, if Zj = 00, then Z - Zj is replaced by l/z. We ask for a criterion matrix for this problem. Note that if mj > 0 then the order mj of pole of f at Zj and the first nj coefficients CjO,··· ,Cjn;-l of the Laurent expansion of f at Zj are prescribed. For this purpose, as in the preceding section §4, we consider a conformal mapping

t.p : D

--+

Do

of D onto the open unit disc Do, the linear fractional mapping defined by

t/J(w) = _P_, w-a

the function 1

Fo(z, () = 1 _ t.p(z)t.p«() , and the matrices

with appropriate replacement as in §4 if 00 presents. Now, for n E N, we introduce the standard n x n nilpotent matrix

o 1

J

Thkahashi

316

where n is a positive integer. Then Nnn (unit matrix). For each Zi, put

=0

(zero matrix). We define N:!

= In

CiO

n;-l Ci

=

L

Cia

N::';+n;

Cin;-l

=

o

a=O

o Ti = N~~n;' Ri = Ci -aTi

0

Cin;-l··· CiO

(i=1,···,k).

If mi > 0 then the diagonal entries of the triangular matrix R; are all equal to CiO, which is not zero by assumption, and hence Ri is invertible. If mi = 0, we may assume for a moment as in §4 that CiO :f: a, so that Ri is invertible. This assumption can be removed as in the final part of §4. A meromorphic function f in D with values in W is transformed by 'IjJ into a holomorphic function p

g(z) = f(z) - a in D with

Igl :5 1.

On the other hand, writing

f(z)

=

fo(z) (z - Zi)m;

and ~z)=

p(z - Zi)m;

fo(z) - a(z -

we see easily that the conditions (EI) for

Zi)m;

,

f are transformed into the conditions

ni-l

(EI)# g(z) = (z -

Zi)m;

(L dia(Z -

+ O«z -

Zi)a

Zi)n;»)

(i = 1,··· , k),

a=O

where, if Zi = 00, then defined by the relations

Z - Zi

is repla.ced by liz and the coefficients

ni-l

" m/+a L..Jdia N m;+n; a=O

-

R-1T,. P i'·

dia

are

317

Takahashi

Denoting this matrix pR";lTi by Di and setting

and

we observe by Theorem 7 that the criterion matrix for the problem (EI)# in 8 is A # and we have

RiA~R; = (Ci - aTi)rij(C; - aTn - p2TirijTj*

= CirijC; -

aTirijC; - aCirijTj

Write

C=

T=

[T'

and define

p2)Ti r ijTj.

r~ [r,:.. r" 1

cJ J

[C'

+ (lal 2 -

rkl ... ru

R=

A = crc* - aTrc* - acrT*

J

[ R,

+ (lal 2 -

p2)Tr T*.

It should be noted that W is expressed by

Then we have A = R A# R*, where R is an invertible matrix. It follows that A ~ 0 if and only if A # ~ 0 and that rank A = rank A #. Theorem 7 yields thus THEOREM 8. Let the notations and the assumption be as above. There exists a meromorphic function f in D with values in W, which satisfies the conditions

1

(EI)

f(z) = (

n;-l

). Z -

Zi m,

(L Cia(Z a=O

Zi)a

+ O«z -

Zit;»)

(i=l,···,k)

if and only if the Hermitian matrix A is positive semidefinite. Such a function is unique if and only if A ~ 0 and det A = o.

318

Takahashi

PART

III. DOMAINS OF FINITE CONNECTIVITY

Let D be a bounded domain in the complex plane whose boundary aD consists of m + 1 pairwise disjoint analytic simple closed curves 'Yi (i = 0,1,··· ,m). In this part, we generalize the results of Abrahamse [1] on the Pick interpolation problem in D, replacing this problem by our extended interpolation problem (EI) and introducing appropriate matrices. The proof proceeds as that of Abrahamse. We point out that this Part III gives another proof of the main theorem of the Part II in the unit disc. §6. Preliminaries. We consider the harmonic measure dw on aD for the domain D and for a fixed point z* ED. In terms of dw, we define the norm II/lIp (1 $ P $ 00) of complexvalued measurable functions I on aD :

1I/IIp =

(rlaD I/I

11/1100 =

ess.sup III

P

dw )

aD

~

(l$p.1,··· ,Am) : Ai E C, IA;! = 1 (i = 1,··· ,m)} be the m-torus. In order to clarify the basic branch of multiple-valued modulus automorphic functions in D, we consider as in Abrahamse [1] m pairwise disjoint analytic cuts OJ (i = 1,· .. ,m), which stapts from a point of 'Yi and terminates at a point of 'Yo. The domain Do = D \ (U~l Oi) is thus simply connected. For A = (At,.·. ,Am) E A, let H). (D) denote the set of complex-valued functions I in D such that I is holomorphic in Do and that, for each t E OJ n D, I(z) tends to I(t) when z E Do tends to t from the left side of Oi and I(z) tends to A;f(t) when z E Do tends to t from the right side of 8;. We can easily verify that if one miltiplies by Ail the values of I on the right side of Oi then the function thus modified is holomorphic at every point of Oi n D. We define the canonical function V). in H).(D) in the usual following way: For each i = 1,··· ,m, let Vi be the harmonic function in the neighborhood of D = DUaD such that Vi = Ion 'Yi and Vi = 0 on the other 'Yj (j =f. i, 0 $ j $ m), and let iT; be the conjugate harmonic function of Vi in Do. For t E OJ n D (j = 0,··· ,m), Vi(t) stands for the limit of iT;(z) when Z E Do tends to t from the left side of OJ. There exist m real numbers 6,··· such that

,em

V).(Z) =

exp(Eej(vj(z) + i Vj(z») j=l

Thkahashi

319

belongs to H~(D) (see Widom [16]). We see that V~ can be continued analytically across the boundary aD, in the usual sense except at the end points of the cuts Di and in the following sense at an end point t of Di : multiplying by A;l the values of V~ on the right side of Di, one can continue analytically across aD the function thus modified in the neighborhood of t. For a function 1 in D, 1 E H~(D) if and only if 1 V~-l is holomorphic in D. Clearly, IV~I is single-valued in D and can be extended to a continuous function in a neighborhood of D, which has no zeros there. Let H~ denote the space of all functions 1 in H~(D) such that 1/12 :::; u in D for some harmonic function u in D. It is known that any function 1 in H~ admits nontangentiallimits /*(t) at almost all t E aD (w.r.t.dw). Via 1 1-+ /*, the space H~ can be viewed as a closed subspace of the Hilbert space L2. Thus H~ is a Hilbert space with the inner product

(I, g)

=f

(f,g E H~).

laD /*g* dw

From now on, no distinction will be made between a function 1 in H~ and its boundary function /* in L2. If A = (1"" ,1) is the identity of A, then H~(D) is the space of holomorphic functions in D and H~ is the usual Hardy space H2(D). It is easy to see that, for A E A and ( E D, the mapping

1 1-+ I«()

is a bounded linear functional on H~, so that we have a unique k~, E H~ such that (for every 1 E

Hn.

We write and The properties of k~ in the following Lemma 1 are known (see Widom [16]). LEMMA 1. For A E

A, zED and (E D, we have

When A u~ed, k~(z,() u holomorphic w.r.t. (z,() in Do xDo. It u continuous on A x Do x Do a" a function 01 (A, z, (). The function Ik~(z, ()I i" "ingle-valued and continuow on A x «D xD)U(D x D)) with it& appropriate boundary values.

320

Takahashi

LEMMA 2. Let A be fixed in A. For to E {JD and (0 E D, there exist a neighborhood U1 of to and a neighborhood of U2 of (0 in D such that the function VA(z)-lk A(z,() VA «() -1 can be extended to a function holomorphic w.r.t. (z,() in U1 x U2 •

Roughly speaking, Lemma 2 says that k>.(z, 0 can be continued analytically across the boundary as a function of two variables (z, (). The presence of VA is only to simplify the statement concerning the cuts hi. This Lemma 2 seems to be well known, but we could not find it in an explicit form in the bibliographies, so that we shall give its proof later in §8. Let 0, f3 be nonnegative integers. For a holomorphic function I( z) we shall denote the o-th derivative of f by 1(01). For a function F(z, () holomorphic w.r.t.

(z, (), the notation F(OI,P) will stand for {JOIH ~, although this is a slight abuse {JZOl{J( of the notation. Let A be fixed in A. It is obvious that the derivative k~OI,P)(Z, 0 is well defined and holomorphic w.r.t. (z,() in Do x Do. By Lemma 2, the function VA(Z)-lk~OI,P)(z,O VA (O-1 can be extended to a function holomorphic w.r.t. (z, () in a neighborhood of (D x D)U(D x D). For t E hi and ( E Do, k~OI,P)(t, () is defined to be the limit of k~OI,P)(Z, 0 when z E Do tends to t from the right side of hi, and so on. The function Ik~Q,P)(z, 01 can be considered as a function single-valued and continuous on (D x D) U (D x D). Though the following lemmas are valid for the points on the cuts, we shall restrict ourselves to Do in order to simplify the statement. This will be sufficient to apply them later. LEMMA 3.

For ( E Do, k~OI,P)( , () belongs to H~ n LOO. We have

and

(z E Do).

In fact, the first assertion is obvious. Since kiO,OI)(t, () = :(: kA(t, () is bounded on ({JD \ D, we have

U:'1 hi) x (Do n D') for any relatively compact domain D'

in

321

Takahashi

and the second equality, replacing

I

by kiO,P)( ,e).

We denote by HOO(D) the space of bounded holomorphic functions in D and we regard it as a closed subspace of Loo. LEMMA

4. Let I E Hoo(D). Put

F(z,e)

= l(z)k>.(z,e)/(e)

and

G(z,O = k>.(z,e)/«()·

Then F and G are holomorphic w.r.t. (z, () in Do x Do and G(O,P)( ,e) belongs to H~ lor any e E Do. We lave (z E Do, e E Do) . In fact, we have

and

(G(O,P)( ,(), G(o,a)( ,z») =

~t

(:)

(~)/(a-I')(z)/(P-I)(O(kiO'")( ,e), kiO'I')( ,z»).

Lemma 3 shows Lemma 4. Let P>. denote the orthogonal projection of L2 onto HI. LEMMA

5. Let I

E

HOO(D) and put G(z,e) = k>.(z,e)/(O. Then we have P>.(/ kio,a)( ,e» = G(o,a)(

,0.

To prove it, let cp E H~. By Lemma 3 we observe

( cp, p>.(/kio,a) (

,(») = (cp, jkio,a) ( ,e»)

= (/cp, kio,a)( ,e») = =

(fcp)(a)(o

~ (:)J(cp, klO,;')( ,0)

= (cp, ~ (:)/(a-I)(e)klO,")( = (cp, G(O,a)( ,e») ,

,e»)

Takahashi

322

which shows Lemma 5. We shall denote by SJ. the orthogonal complement of a subspace S in L2. LEMMA 6 (ABRAHAMSE [1]). Let w be an invertible function in L=. If f

is in (w H2(D»J. such that

n L= then there exist some ..\

E A, 9 E H~ and h E (w H~)J.

(a.e. on aD).

and

LEMMA

7 (ABRAHAMSE [1]). The linear subspace (H2)J. n L= is dense in

(H2)J..

§7. Main Theorem in Finitely Connected Domains. Let z}, Z2,' •. , Zk be k distinct points in a domain D in C, bounded by m + 1 disjoint analytic simple closed curves. For each Zi, let CiO,'" ,Cin;-l be a sequence of ni complex numbers. Our present extended interpolation problem (EI) is to find a holomorphic function f in D, satisfying If I :5 1 and the conditions ni-l

(EI)

fez) =

L

Ci(Z -

Zi)

+ O«z -

Zit;)

(i=I, .. ·,k).

=0

k>.

For each element ..\ of the m-torus A, let be the kernel function introduced in the preceding section. We define the following matrices for i, j = 1" .. ,k: CiO

Gi=

[

Cil

CiO

Cin~-l A~~) = r~~) '}

A>. =

'}

-

Gi . r~~) . G}~, '}

[Ai~: .. ~ '. '... ~~~) ]. A~)

...

A~~

In tenus of these matrices A). (..\ E A), we have

Takahashi

323

THEOREM 9 (EXTENSION OF TilE THEOREM OF ABRAHAMSE). The problem (EI) admi~ a solution f with If I :5 1 in D if and only if the matrix A,\ is positive semidefinite for each oX in A. The solution is unique if and only if the determinant of A,\ is zero for some oX E A. PROOF. We may assume without loss Zi E Do = D \ (U:: o 6i). Let eia be complex numbers (i = 1,'" , k ; a = 0"" , ni - 1) and set k

K =

(1)

n;-l

LL

eiak\o,a)( , Zi).

i=l a=O By Lemma 3, we have k

IIKII~ =

n;-l nj-1

L L L

eiaejp kia,P)(Zi, Zj).

i,j=l a=O p=o Let

f E HOO(D). Put, for z E Do, ( E Do, F(z,()

= f(z)k,\(z,()f«()

and

G(z,()

= k,\(z,()f«().

By Lemma 5, k

P,\(lK) =

n;-l

L L eiaG(O,a)( ,Zi). i=l 0=0

It follows from Lemma 4 that k

IIP>.(lK)II~ =

n;-l nj-1

L L L

eioejpF(o,P)(Zi, Zj).

i,j=l a=O p=o On the other hand, we have

and hence k

(2)

n;-l nj-1

L L L

eiaejp(lIfll~ kio,P)(Zi,Zj) - F(o,P)(Zi,Zj» ~ O.

i,j=l 0=0 p=o Now, assume that f satisfies If I :5 1 and the conditions (EI). Then, by the product formula (PF) in §1 and the definition of F, we see that kio,lI)(Zi,Zj)-

Thkahashi

324

F(Ot,{J)(Zi, Zj) is the (a+1, !3+1)-entry of the matrix of ;t~;). This implies A~ ~ 0 for each A E A. To prove the converse, assume A~ ~ 0 for each A E A and take a polynomial 4>( z) which satisfies the conditions (EI). We may find such a polynomial by the method of indeterminate coefficients. Let w( z) be the polynomial w(z) = (z - Zt}Rl ••• (z - Zk)R~. It is easy to see that the subspace wHi is the orthogonal complement in Hi of the subspace M ~ spanned by the functions {klO,Ot)( ,Zi)} (i = 1"" ,k; a = n," . ,ni - 1). Thus we have

Hi

= M~

ffi wHi·

Let/beinVlOn(wH2(D».L. ByLemma6,thereexistsomeA E A, 9 E Hi and h E (w Hi).L such that

/=gh

(a.e. on aD).

and

The function 1( = P~(h) is in M~ and hence it can be written in the form (1). Since A~ ~ 0, we have IIP~(~1()1I2 ~ 111(112 by (2). Hence,

IlaD ~/ dwl = IlaD ~9hdwl = 1< h, 4>g >1 =

1< 1(, 4>g >1 = 1< P~(~1(), 9 >1

~ IIP~(~1()1I2I1gI12

~ IIKII211g112 ~ IIhll2 11g112 = II/Ill.

By the theorem of Hahn-Banach, there exists a function tP in Loo such that IItPlioo ~ 1 and that for each / E Loo n (WH2(D».L we have

f tij/dw= f ~/dw. laD laD This shows, by virtue of Lemma 7 and the relations H2 (D) = M ~1 ffi wH2(D) and M~l c Hoo(D), where Al is the identity of A, that tP - if> is orthogonal to (wH2(D».L in L2, that is, it belongs to wH2(D). Therefore, tP is a solution of the problem (EI) with ItPl ~ 1, which completes the proof of the first part of the theorem. To prove the uniqueness assertion, it suffices to follow the proof of Abrahamse [1], using instead of his Lemma 6 in [1] the following lemma which will be deduced immediately from Lemma 1 and Cauchy's integral formula. The details will not be carried out here. LEMMA 8. Let (zo, (0) be in Do x Do. Let a And fJ be two nonnegative integer,. Then the mapping A ....... kiOt,{J)(zo,(o) i, continuo,,", On the m-torw A.

325

Takahashi

§8. Proof of Lemma 2 It is known that, for a fixed (, the function k),(z, 0 of z can be continued across the boundary aD. The problem is to find, for a relatively compact neighborhood U2 of (0 in D, a connected neighborhood U1 of to common to all ( E U2 such that, multiplying if necessary by .A;1 the values on the right side of the cut 6;, we may continue the function V),(z) to a function holomorphic and invertible in U1 and that, for any fixed ( E U2 , the function V),(z)-l k),(z, 0 V),(O -1 of z may be extended to a function holomorphic in U1 • If we find such a neighborhood U1 , then it will follow from the theorem of Hartogs [8] that the function thus extended to U1 for each (E U2 is holomorphic w.r.t. (z,() in U1 x U2, since the original function is holomorphic w.r.t. (z, () in (U1 nD) x U2 • This will complete the proof of Lemma 2.

Now, we reduce by means of V), to the case without the periods .A but with a slightly modified measure m

(3)

dtt(t)

= eXP(L 2~;v;(t»)

dw(t).

;=1

The kernel function k(z, 0 of H2(D) w.r.t. dtt, which satisfies by definition

(4)

J(O =

f J(t) k(t, 0 JaD

dtt(t)

has the relation

k),(z,O

= V),(z) k(z,O V),(O

(see Widom [16]), so that it suffices to prove the Lemma 2 for k(z, O. Let g(z, z*) be the Green function of D with its pole at the reference point z* and let g(z,z*) be its harmonic cunjugate. Put G(z) = g(z,z*) + i g(z,z*). Then we have

dw(t) = ~G'(t) dt . 271"

(5)

The function G' is single-valued and holomorphic in D except at the single pole z*. It can be continued analytically across the boundary aD by virtue of the reflection principle. It has m zeros zi,· .. ,z;' in D but it does not vanish on

aD. For

J E H2(D) we have J(C) = ~ ( J(t) dt 271"z JaD t - (

«( E D)

Takahashi

326

(see Rudin [12]), so that (4) yields

f f(t) laD

(1 t _1( - -k(t, () ----;It dP(t») dt 271"i

= 0

This shows that there exists a unique N E HOO(D) such that

(6)

N(t) = ~_1__ k(t,() dp(t) 271"tt - ( dt

(t E aD)

(see Rudin [12]). The function 1

M(z) = - . -

1

271"1 Z -

(

- N(z)

is holomorphic in D except at the single pole (. Assume to E Ij. As exp(E:: 1 2eiVi(t») is a constant (3), (5), and (6)

k(t, () =

Cj

·"M(t) . G'(t)-l

=f. 0 on Ii> we have by

(t E Ij),

where Cj is a constant =f. O. The function Lj = CjMG,-l is meromorphic in D and its poles are at most at (, zi, ... Since P = Lj + k( ,() is real and Q = Lj - k( ,() is purely imaginary on Ij, the functions k( ,() and Lj can be cuntinued analytically across Ij by the reflection principle as well as P and Q. Let U2 be a relatively compact neighborhood of (0 in D. Since the zi are independent of (, we can find a neighborhood U1 of to, which is symmetric w.r.t. Ij and contains no or no points of U2 • Then P and Q, and hence Lj and k( ,(), can be extended to holomorphic functions of z in U1 for any ( E U2 , which completes the proof of Lemma 2.

,z:..

z;

References

[I] M. B.Abrahamse, The Pick interpolation theorem for finitely connected domains, Michigan Math. J. 26 (1979), 195-203.

[2] L. Ahlfors, Conformal Invariants. Topics in Geometric Function Theory, McGraw-Hill, New York, 1973. [3] C. Carathoodory, Uber den Variabilitii.tsbereich der Fourier'schen Konstanten von positiven harmonischen Funktionen, Rend. Circ. Mat. Palermo 32 (1911), 193-217. [4] J. P. Earl, A note on bounded interpolation in the unit disc, J. London Math. Soc. (2) 13 (1976), 419-423.

Takahashi

327

[5] S. D. Fisher, FUnction Theory on Planer Domains, Wiley, New York, 1983. [6] P. R. Garabedian, Schwarz's lemma and the Szego kernel function, Trans. Amer. Math. Soc. 67 (1949), 1-35. [7] J. B. Garnett, Bounded Analytic Functions, Academic Press, New York, 198!. [8] F. Hartogs, Zur Theorie der analytischen Funktionen mehrerer unabhiingiger Veriinderlichen, insbesondere iiber die Darstellung derserben durch Reihen, welche nach Potenzen einer Veriinderlichen fortschreiten, Math. Ann. 62 (1906), 1-88. [9] D. E. Marshall, An elementary proof of Pick-Nevanlinna interpolation theorem, Michigan Math. J. 21 (1974), 219-223. [10] R. Nevanlinna, tiber beschriinkte analytische Funktionen, Ann. Acad. Sci. Fenn. Ser A 32 (1929), No 7. [11] G. Pick, tiber die Beschriinkungen analytischer Funktionen, welche durch vorgegebene Funktionswerte bewirkt werden, Math. Ann. 77 (1916), 7-23. [12] W. Rudin, Analytic functions of class H p , Trans. Amer. Math. Soc. 78 (1955), 46-66. [13] I. Schur, tiber Potenzreihen, die im Innern des Einheitskreises beschriinkt sind, J. Reine Angew. Math. 147 (1917), 205-232. [14] S. Takahashi, Extension of the theorems of Caratheodory-Toeplitz-Schur and Pick, Pacific J. Math. 138 (1989), 391-399. [15] S. Takahashi, Nevanlinna parametrizations for the extended interpolation problem, Pacific J. Math. 146 (1990), 115-129. [16] H. Widom, Extremal polynomials associated with a system of curves in the complex plane, Advances in Math. 3 (1969), 127-232.

Department of Mathematics Nara Women's University Nara 630, Japan

MSC 1991: Primary 30E05, 30C40 Secondary 47A57

Operator Theory: Advances and Applications, Vol. 59 © 1992 Birkhiiuser Verlag Basel

328

ACCRETIVE EXTENSIONS AND PROBLEMS ON THE STIELTJES OPERATOR-VALUED FUNCTIONS RELATIONS

E. R. Tsekanovskii

Dedicated to the memory of M. s. Brodskii, M. G. Krein, S. A. Orlov, V. P. Potapov and Yu. L. Shmul'yan This paper presents a survey of investigations in the theory of accretive extensions of positive operators and connection with the problem of realization of a Stieltjes-type operator-valued function as a linear fractional transformation of the transfer operatorfunction of a conservative system. We give criteria of existence, together with some properties and a complete description of a positive operator.

In this paper a survey of investigations in the theory of accretive extensions of positive operators and connections with the problem of realization of a Stieltjes-type operator-valued function as a linear fractional transformation of the transfer operatorfunction of a conservative system is given. We give criteria of existence together with some properties and a complete description of the maximal accretive (m-accretive) nonselfadjoint extensions of a positive operator with dense domain in a Hilbert space. In the class of m-accretive extensions we specialize to the subclass of 8-sectorial extensions in the sense of T. Kato [24] (in S. G. Krein [29] terminology is regularly dissipative extensions of the positive operator), establish criteria for the existence of such extensions and give (in terms of parametric representations of 8-cosectorial contracitive extensions of Hermitian contraction) their complete description. It was an unexpected fact that if a positive operator B has a nonselfadjoint m-accretive extension T (B eTc B*) then the operator B always has an m-accretive extension which is not 8-sectorial for any 8 (8 E (0,11"/2». For Sturm-Liouville operators on the positive semi-axis there are obtained simple formulas which permit one (in terms of boundary parameter) to describe all accretive and 8-sectorial boundary value problems and to find an exact sectorial angle for the given value of the boundary parameter. All Stieltjes operator-functions generated by positive Sturm-Liouville operators are described. We obtain M. S. Livsic triangular models Jor m-acretive extensions (with real spectrum) of the positive operators with finite and equal

1Sekanovskii

329

deficiency indices. In this paper there are also considered direct and inverse problems of the realization theory for Stieltjes operator-functions and their connection with 9-sectorial extensions of the positive operators in rigged Hilbert spaces. Formulas for canonical and generalized resolvents of 9-cosectorial contractive extensions of Hermitian contractions are given. Note that Stieltjes functions have an interesting physical interpretation. As it was established by M. G. Krein any scalar Stieltjes function can be a coefficient of a dynamic pliability of a string (with some mass distribution on it). §1. ACCRETIVE AND SECTORIAL EXTENSIONS or THE POSITIVE OPERATORS, OPERATORS or THE CLASS C(9) AND THEIR PARAMETRIC REPRESENTATION. Let A be a Hermitian contraction defined on the subspace :D(A) of the Hilbert space i). DEFINITION. The operator S E [i),i)] ([i),i)] is the set of all linear bounded operators acting in i)) is called a quasi-selfadjoint contractive extension (qsc-extension) of the operator A if S:::> A, S*:::> A, IISII:5 1

:D(S) = i). Self-adjoint contractive extensions (sc-extensions) of the Hermitian contraction were investigated, at first, by M. G. Krein [27], [28] in connection with the problem of description and uniqueness of the positive selfadjoint extensions of the positive linear operator with dense domain. It was proved by M. G. Krein [27] that there exists two extreme sc-extensions A" and AM of the Hermitian contraction A which are called "rigid" and "soft" sc-extensions of A respectively. Moreover, the set of all sc-extensions of the Hermitian contraction A consists of the operator segment [A", AM]. Let B be a positive closed Hermitian operator with dense domain acting on the Hilbert space i). Then, as i~ is well known [27], an operator A = (1 - B)(1 + B)-1 is a Hermitian contraction in S) defined on some subspace :D(A) of the Hilbert space i). Let A" and AM be "rigid" and "soft" sc-extensions of A. Then (see [27]) the operator B" = (1 - A,,)(1 + A,,)-1 is a positive self-adjoint extension of B and is in fact the extension obtained, at first, by K. Friedrichs in connection with his theorem that any positive operator with dense domain always has a positive self-adjoint extension. Besides, as it proved was in [27], the operator 8M = (1 - AM)(1 + AM)-1 is also a positive self-adjoint extension of B. We call the ,perators B" and BM the K. Friedrichs extension and the M. Krein extension respectively. DEFINITION. Let T be a closed linear operator with dense domain acting on 'he Hilbert space i). We call T accretive if fu!(TI,f) ~ 0 (VI E :D(T» and m-accretive if (4 does not have accretive extensions.

lSekanovskii

330

DEFINITION. We call the m-accretive operator T 8-sectorial if there exists

8 E (0, 1r /2) such that ctg 8 IIm(TI,f)1 $ Re(TI,f)

(VI

E ~(T».

(1)

DEFINITION. An operator S E [fj,fj] is called a 8-cosectorial contraction if there exists 8 E (0, 1r /2) such that

2ctg8IIm(SI,f)1 $

11/112 -

IISIII 2

(VI

E fj).

(2)

Note that inequality (2) is equivalent to (3)

Denote by 0(8) the set of contractions S E [fj,fj] (8 E (0,1r/2» satisfying (2) (or (3)) and let 0(1r/2) be the set of all self-adjoint contractions acting on fj. It is known [43] that if T is a 8-sectorial operator, then the operator S = (1 - T)(1 + T)-1 is a 8-contraction, i.e. S belongs to 0(8). The converse statement is also valid, i.e. S E 0(8) and (1 + S) is invertible, then the operator T = (1 - S)(1 + S)-1 is a 8-sectorial operator. THEOREM 1. (E. R. Tsekanovskii [40), [41]) Let B be a positive linear closed operator with dense domain acting on the Hilbert space fj. Then the operator B has a nonselfadjoint m-accretive extension T (B eTc B*) if and only if the K. Friedrichs extension B,. and the M. Krein extension B M do not coincide, i.e. B,. =F B M. If B,. =F B M,

then 1. for every fixed 8 E (0, 1r /2) the operator B has nonselfadjoint 8-sectorial extension T (B eTc B*)j 2. the operator B has a nonselfadjoint m-accretive extension T (B eTc B*) that fails to be 8-sectorial for all 8 E (0, 1r /2).

The description of all 8-sectorial extensions of the positive operator B can be obtained via a linear fractional transformation with the help of the following theorem on parametric representation. THEOREM 2. (Yu. M. Arlinskii, E. R. Tsekanovskii [7J, [8J, [9]) Let A be a Hermitian contraction defined on the subspace ~(A) of the Hilbert space fj. Then the equality

(4)

331

Tsekanovskii

establishes one-to-one correspondence between qsc-extensions S of the Hennitian operator A and contractions X on the subspace lJlo = 9t(AM - A,,). The operator S in (4) is a 8cosectorial qsc-extension of the Hennitian contraction A iff the operator X is a 8-cosectorial contraction on the subspace lJlo. Let A E [:D(A),fj] be a Hermitian contraction, A* E [fj,:D(A)] be the adjoint of the operator A. Denote 9 = (1 - AA*)1/2:D(A), £. = (1 - AA*)1/2fj e 9 and lJl = fj e :D(A). Let P A, P IJI , Pc. be orthoprojectors onto :D(A), lJl, £. respectively. We will define the contraction Z E [9, lJl] in the form Z(I - AA*)1/2 f = PIJIAf

(J E :D(A»

and let Z* E [lJl, 9] be the adjoint of the operator Z.

THEOREM 3. Let A be a Hennitian contraction in fj defined on the subspace :D(A). Then the equalities

+ (1 APA + (1 -

AI' = APA AM =

are valid. The equality AI'

AA*)1/2(Z* PIJI - Pd1 - AA*?/2) AA*)1/2(Z* PIJI

+ Pd1 -

AA*)1/2)

= AM holds if and only if £. = {OJ.

Theorem 3 was established by Yu. M. Arlinskii and E. R. Tsekanovskii [8]. In terms of the operator-matrix "rigid" and "soft" extensions were established by Azizov [1] and independently by M. M. Malamud and V. Kolmanovich [25]. The M. Krein extension BM of the positive operator B with dense domain was described at first by T. Ando and K. Nishio [4], later by A. V. Shtraus [39]. In terms of the operator-valued Weyl functions [17] and space of boundary values operator BM was described by V. A. Derkach, M. M. Malamud and E. R. Tsekanovskii [18]. Contractive extensions of the given contrac;ion in terms of operator-matrices were investigated by C. Davis, W. Kahan and H. Wein>erger [16], M. Crandall [15], G. Arsene and A. Gheondea [5], H. Langer and B. Textorius [34], Yu. Shmul'yan and R. Yanovskaya [38]. Theorems 1 and 2 develop and reinforce investigations by K. Friedrichs, M. G. Krein, R. Phillips [27], [28], [37] and give the solution of the T. Kato problem [24] on the existence and description of 8-sectorial extensions of the positive operator with dense domain. Note that m-accretivity (8-sectoriality) of an operator T is equivalent to the fact that the solution of the Cauchy problem f

dx { -+Tx=O dt x(O) = Xo

(xo

E :D(T»

1Sekanovskii

332

generates a contractive semigroup (a semigroup analytically continued as a semigroup of contractions into a sector Iarg el < 7r /2 - 8 of complex plane) (29). Now, we consider some applications of Theorem 2 to the parametric representation of the 8-cosectorial qsc-extensions of a Hermitian contraction. Let S be a linear bounded operator with finite dimensional imaginary part acting on the Hilbert space fj. Then as it is known (32) r

ImS =

L

(-,ga)ia{Jg{J

a,{J=l where J = [ja{J) is a self-adjoint and unitary matrix. Consider the matrix function YeA) given by

THEOREM 4. In order that the linear bounded operator S with finite dimensional imaginary part be a contraction, it is necessary and, for simple 1 S, sufficient that the following conditions are fulfilled: 1) yeA) is holomorphic in Ext[-I, 1) 2) the matrices V-1( -1) = (V( -1 - 0))-1, V-1(I) = (V(I + 0))-1, (V-1( -1) V- 1(I))-1/2 exist and the matrix KJ

= (V-1( -1) -

V-1(I))-1/2{2iJ + V- 1( -1) + V- 1(1)} x (V- 1 ( -1) _ V- 1(I))-1/2

(5)

is a contraction. The contraction S is 8-cosectorial (belongs to the class C( 8), 8 E (0, 7r /2)) iff K J of the mrm (5) is a 8-cosectorial contraction (belongs to the class C( 8), 8 E (0, 7r /2)). Moreover the exact value of the angle 8 is defined from the equation

This theorem was obtained by E. R. Tsekanovskii (42), (43). For the operator with one-dimensional imaginary part another proof was given by V. A. Derkach (20).

EXAMPLE. Let a(x) nondecreasing function on [O,l). We consider the operator (Sf)(x) = a(x)f(x) + i

it

f(t)dt

acting on the Hilbert space L2[O,£). It is easy to see that (ImSf)(x)

1

rt f(t)dt = (f,g)g

= '210

(g(x)

== I/~)

(6)

lThe operator S is called simple if there exists no reducing subspace on which one induces a self-adjoint operator.

Tsekanovskii

333

(ReSf)(x) = a(x)f(x) + ~

it

f(t)dt -

~ 1% f(t)dt.

From simple calculations

1 It dt ) V(A) = «ReS - M)-lg,g) = tg ( 210 a(t) _ A . Set a( x) ==

°

(7)

and consider the operator

(Sof)(x) = i

it

f(t)dt

(x E [O,l], f E L2[0,l]).

(8)

From (6) and (7) it follows that V(l) = -tg(l/2), V(-l) = tg(l/2). As J = I and an operator So is simple, we shall find all l > for which So is a 8-cosectorial contraction. For this, in accordance with Theorem 4 (see relation (5», the number 2i + V- 1 (1) + V- 1(-1) K = KJ = V-l( -1) _ V-l(l)

°

has to satisfy the inequality

Exact value of 8 can be calculated from the formula

1-IK12 ctg8= 2IImKI.

°

The operator So is a 8-cosectorial contraction if and only if < l < 11"/2, moreover, an exact value of 8 is equal to l (8 = i). From this and the definition of the class C(8) (8 E (0,11"/2», we obtain the inequality

ctg'll,'

J(t) dt!' ,;

I,'

IJ(t)I'dt -

I,' If.'

J( t) dt!' d%

(Vi E [0,11"/2], Vf E L 2 [0,i]). Moreover, the constant ctgl can not be increased so that for all f E L2[0,l] as mentioned 9.bove, the inequality is valid. With the help of Theorem 4 there was established a full description of positive And sectorial boundary value problems for Sturm-Liouville operators on the semi-axis, at first (see also [22], [26]). Let fj = L2[a, 00], l(y) = -y" + q(x)y, where q(x) is a real locally ~ummable function. Assume that a minimal Hermitian operator

{ By = l(y) = -y" + q(x)y y'(a) = y(a) = 0

(9)

1Sekanovskii

334

has deficiency indices (1,1) (the case of limiting Weyl point). Let tt'k(X,A) (k = 1,2) be the solutions of the Cauchy problems

Then, as it is known [35], there exists Weyl function moo(A) such that

Consider a boundary problem {

ThY = l(y) = -y" + q(x)y

(10)

y'(a) = hy(a).

THEOREM 5. (E. R. Tsekanovskii [42], [43]) 1. In order that the positive Stwm-Liouville operator of the form (9) have nonselfadjoint accretive extensions (boundary problems) of the form (10), it is necessary and suJIicient that moo( -0) < 00. 2. The set of all accretive and 8-sectorial boundary value problems for Stwm-Liouville operators of the form (10) is defined by the parameter h, belonging to the domain indicated in (11). Moreover, as 1) h sweeps the real semi-axis in this domain, there results all accretive self-adjoint boundary value problems; 2) h sweeps all values not belonging to the straight line Re h = -m oo ( -0) and h =F h, then there results all 8-sectorial boundary value problems (8 E (0, 7r /2»); moreover, the exact value of 8 is defined as it was pointed out in (11); 3) h sweeps all values with h =F h and belonging to the straight line Re h = -m oo ( -0), then there results all accretive boundary value problems which are not 8-sectorial for any 8 E (0, 7r /2).

h

(11)

1Sekanovskii

335

Thus, Theorem 5 indicates which values of boundary parameter h correspond to the semigroup generated by the solution of the Cauchy problem

dx = 0 { -+Thx dt

x(O) =

Xo

in the space L 2 [0,00], being contractive (Re h ~ -m oo ( -0», and for which h it can be analytically continued as a semigroup of contractions into a sector 1arg (I < 1r /2 - 8 of the complex plane. At the same time, it helps to calculate 8. Also the value of the parameter h in (11) (Re h > -m oo ( -0» determines whether this semigroups of contractions can not be analytically continued as a semigroup of contractions into any sector 1arg (I < e of the complex plane (Reh = -moo(-O» (Imh =F 0). Note that the M. Krein boundary value problem for the minimal positive operator B of the form (9) has the form (as it follows from (11»

{

BMY = -Y" + q(x)y y'(a) + moo( -O)y(a)

=0

(x E [a,ooD

and the K. Friedrichs boundary value problem, as is well known, coincides with Dirichlet problem B"y = -y" + q(x)y { (x E [a,ooD. y(a) = 0 EXAMPLE. Consider a Sturm-Liouville operator with Bessel potential

{

By = -y" +

V2

-1/4 y x2

(x E [1,00], v

~

1/2)

y'(I) = y(I) = 0 in the Hilbert space L2[I, 00]. In this case the Weyl function has the form [35] m (A) = 00

-v'X 1~(VI) - iY~(VI) 1~( v'X) + iY~( VI)

where 1,,(z), Y,,(z) are Bessel functions of the first and second genus, moo( -0) = v. §2. ALIZATION.

STIELTJES OPERATOR-VALUED FUNCTIONS AND THEIR RE-

Let B be a closed Hermitian operator acting on the Hilbert space 1), B* be the adjoint of B, ~(B*) = 1), 9t(B*) C 1)0 = i>(B). Denote 1)+ = i>(B*) and define in 1)+ scalar product (/,g)+ = (/,g) + (B*J,B*g)

1Sekanovskii

336

and build the rigged Hilbert space fj+ C fj C fj_ [14). We call an operator B regular, if an operator PB is a closed operator in fjo (P is an orthoprojector fj to fjo) [6), [46). We say that the closed linear operator T with dense domain in fj is a member of the class 0 B, if 1) T:::> B, T* :::> B, where B is a regular closed Hermitian operator in fj. 2) (-i) is a regular point of T. The condition that (-i) is a regular point in the definition of the class OB is non-essential. It is sufficient to require the existence of some regular point for T. We call an operator A E [fj+,fj+) a biextension of the regular Hermitian operator B if A :::> B, A* :::> B. If A = A*, then A is called a selfadjoint biextension. Note that identifying the space conjugate to fj± with fj'f gives that A* E [fj+,fj_). The operator BI = AI, where ::D(B) = {f E fj+ : AI E fj} is called the quasi-kernel of the operator A. We call selfadjoint biextension A strong if B = B* [45), [46). An operator A E [fj+,fj_) is called a (*)-extension of the operator T in the class OB if A:::> T :::> B, A* :::> T* :::> B where T (T*) is extension of B without exit of the space fjj moreover, A is called correct if An = (A+A*)/2 is a strong selfadjoint biextension of B. The operator T of the class OB will be associated with the class AB if 1) B is a maximal common Hermitian part of T and T*j 2) T has correct (* )-extension. The notion of biextension under the title "generalized extension" , at first, was investigated by E. R. Tsekanovskii [45), [46). (There the author obtained the existence, a parametric representation of (* )-extensions and self-adjoint biextensions of Hermitian operators with dense domain.) The case of biextensions of Hermitian operators with nondense domain was investigated by Yu. M. Arlinskii, Yu. L. Shmul'yan and the author [6), [45), [46). Consider a Sturm-Liouville operator B (minimal, Hermitian) of the form (9) and an operator T" of the form (10). Operators

Ay = -y" + q(x)y +

~h [hy(a) -

y'(a))[Jtc5(x - a) + c5'(x - a»)

A*y = -y" + q(x)y +

~h- [/iy(a) -

y'(a))[Jtc5(x - a) + c5'(x - a»)

JtJt-

(12)

for every Jt E [-00, +00) define correct (* )-extension of T" (T:) and give a full description of these (*)-extensions. Note that c5(x-a) and c5'(x-a) are the c5-function and its derivative respectively. Moreover,

(y,Jtc5(x - a) + c5'(x - a» = Jty(a) - y'(a)

1Sekanovskii

337

DEFINITION. The aggregate e = (A,i)+ C i) C i)_,K, J,e) is called a rigged operator colligation of the class AB if 1) J E [e,e) (e is a Hilbert space), J = 1* = J- 1; 2) K E [e, i)); 3) A is a correct (* )-extension of the operator T of the class A B, moreover, (A A*)j2i = KJK*; 4) 9t(K) = 9t(ImA) + C, where C = i) e i)o and closure is taken in i)_. The operator-function We(z) = I -2iK*(A-zI)-1 KJ is called a M. S. Livsic characteristic operator-function of the colligation e and also M. S. Livsic characteristic operator-function of operator T. The operator colligation is called M. S. BrodskiiM. S. Livsic operator colligation. In the case when T is bounded, we obtain the well-known definition of the characteristic matrix-function [13), [32) (with M. S. Brodskii modification) introduced by M. S. Livsic [32). The other definitions, generally speaking, of unbounded operators were given by A. V. Kuzhel and A. V. Shtraus. For every M. S. Brodskii" M. S. Livsic operator colligation we define an operator-function

Ve(z) = K*(AR - zI)-1 K.

(14)

The operator-functions Ve(z) and We(z) are connected by relations

Ve(z) = i[We(z) + 1)-I[We(z) - 1)J We(z) = [I + iVe(z)J)-I[I - iVe(z)J).

(15)

The conservative system of the form

(A - zI)x = KJ 0) depending on th, 82 • Since 8I, 82 E R, actually T E iR. The point u varies in the period parallelogram, with vertices 0,1, T, 1 + T, say. A complete set of non-equivalent determinantal representations of X is given by d + Y2 q + Yl ) e + Yl 0 , q + Yl 0 -1 _ 81 + 82 - e _ 8 8 _ 81 + 82 - e 81 q2 ,p - 1 2 2 U=

f

f

p ( -d + Y2

+ 82 + 3e 2

= ±l,e E R,d E iR,~ = -e(-e + (1 )( -e + ( 2 )

(2.3)

Note that (-e, -d) is an affine point on X. Choosing a suitable homology basis, .the period lattice A of X in C is spanned by 1, T, and J(X) = C/(Z + TZ) is simply the period parallelogram with opposite sides identified (and is isomorphic to X itself by (2.2». The point ( corresponding to the representation U of (2.3) under the correspondance of Theorem 2.1 is ( = v + 1¥, where v is the point in the period parallelogram corresponding to the point (-e, -d) on X under the parametrization (2.2). The condition 8( () -:f:. 0 gives

v

~

0 (mod 1,T)

(2.4)

which is equivalent to ( -e, -d) being an affine point on X, and (

v

+v= 0

(mod 1,T)

+(

= 0 gives

(2.5)

which is equivalent to e E R, d E iR, in accordance with (2.3). So for smooth cubics Theorem 2.1 is the correspondance between non-equivalent determinantal representations (2.3) and points v in the period parallelogram satisfying (2.4)-(2.5). The set of such points consists of two connected components - a circle To and a punctured circle Tl (see Fig. 2.1). The sign of the representation (2.3) is f. It can be shown that a determinantal representation corresponds to a point v in To if and only if its coefficient matrices admit a positive definite real linear combination, i.e. after a suitable real projective change of coordinates we have, say, U2 > o. T

~------~------~

1/2

Figure 2.1: {v: v

1

t= O,v +11 == O} = ToUTl

Vinnikov

3

353

Commutative Operator Colligations

Operator colligations (or nodes) fonn a natural framework for the study of nonselfadjoint operators. We first recall (see [10]) the basic definition of a colligation for two operators. A colligation is a set

(3.1) where Al> A2 are linear bounded operators in a Hilbert space H, ~ is a linear bounded mapping from H to a Hilbert space E, and 17t, 172 are bounded selfadjoint operators in E, such that (3.2) We shall always assume that dim E = n < 00, while dim H = N $ 00. We shall also assume that ker171 n ker172 = o. A colligation is called commutative if AIA2 = A 2A I . A colligation is called strict if ~(H) = E. If At, A2 are commuting operators with finite-dimensional imaginary parts in a Hilbert space H, then

(3.3) where G = (AI - Ai)H + (A2 - A;)H is the nonhermitian subspace, Pa is the orthogonal projection onto G, and 17k = teAk - A k) I G (k= 1,2). So the pair Ab A2 can always be embedded in a strict commutative colligation with E = G, ~ = Pa. If C = (At, A 2 , H,~, E, 17t, (72) is a strict commutative colligation, there exist selfadjoint operators ,in, ,out in E such that ;'(AIA; - A 2An = z ;'(A;A 1

z

-

~*,in~,

A~A2) = ~*,out~

(3.4)

As evidenced in Section 1, the operators ,in, ,out play an important role, but the condition ~(H) = E is too restrictive. The elementary objects - colligations with dimH = 1 are not strict when dim E > 1. We introduce therefore the notion of a regular colligation

[8,11). A commutative colligation C = (Al> A 2 , H,~, E, 17b (72) is called regular if there exist selfadjoint operators ,in, ,out in E such that

I7I~A; - 172~A; = ,in~, I7I~A2 -172~AI = '"f.0ut~, ,out

=,in + i(I7I~~*172 -172~~*(71)

(3.5)

Actually, it is enough to require the existence of one of the operators ,in, ,out, the other one can then be defined by the last of the equations (3.4) and will satisfy the

Vinnikov

354

corresponding relation. Strict colligations are regular. For a strict colligation the operators 'Yin, 'Y0ut are defined uniquely, but for a general regular colligation they are not, so we will include them in the notation of a regular colligation and write such a colligation as

C = (At,A2,H,~,E'0"1'0"2''Yin"out) Regular commutative colligations turn out to be the proper object to study in the theory of commuting nonselfadjoint operators. As in Section 1 we define the discriminant polynomial of the regular colligation C

(3.6) and (assuming f(yt, Y2) ¢. 0) the discriminant curve X determined by it in the complex projective plane. We have again the Cayley-Hamilton theorem

(3.7) where ff = span {A~' A~2~"(E)}~.k2=o = span {Aik, A;k2~·(E)}k:.k2=o is the principal subspace of the colligat ion , so that the joint spectrum of the operators At, A2 (restricted to the principal subspace) lies on the discriminant curve. Finally,

(3.8) so that the discriminant curve comes equipped with the input and the output determinantal representations. We formulate now the inverse problem of Section 1 in the framework of regular commutative colligations. We are given a real projective plane curve X of degree n, a determinantal representation YI 0"2-Y20"1 +, of X, and a subset S of affine points of X, which is closed and bounded in e 2 , and all of whose accumulation points are real points of X. We want to construct (up to the unitary equivalence on the principal subspace) all regular commutative colligations with discriminant curve X, input determinantal representation !/l0"2 - !/20"l +" and operators At,A2 in the colligation having joint spectrum S (since O"t, 0"2, 'Y are given as n x n hermitian matrices we identify the space E in the colligation with en). It is easily seen that the solutions of this problem that are strict colligations give the solution to the original problem of Section 1 (up to the equivalence of determinantal representations) . Our solution of the inverse problem will be based on a "spectral synthesis", using the coupling procedure to produce more complicated colligations out of simpler ones. Let

be two colligations. We define their coupling

C

= C t V C" = (All A 2 , H, C), E,O"t, 0"2)

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355

where

~,)

(k = 1,2), (3.9)

(the operators being written in the block form with respect to the orthogonal decomposition H = H' EI1 H"). It is immediately seen that C is indeed a colligation. However, if C' and C" are commutative, C in general is not. Assume now C', C" are regular commutative colligations:

0' ",,,in ",,,out) C" -- (A"l ' A"2' H" , ~"E , ,0'1, 2, I "

TheoreIn 3.1 (Livsic [11], Kravitsky [8]) The coupling

where AI, A 2 , H, ~ are given by (9.8), "{in colligation if and only if ,,{,out = "{,,in.

= ,,{,in, "{out = "{,,out,

is a regular commutative

This theorem illustrates aptly the meaning of the input and the output determinantal representations. Note that H" CHis a common invariant subspace of AI, A 2 • Conversely, if H" cHis a common invariant subspace of the operators AI, A2 in a regular commutative colligation C, we can write C = C'V C", where C', C" are regular commutative colligations called the projections of C onto the subspaces H = He H", H" respectively [11,8].

4

Construction of I Triangular Dimensional Case

Models:

Finite-

We shall start the solution of the inverse problem for regular commutative colligations by .investigating the simplest case when dim H = 1 and the joint spectrum consists of a single (non-real) point. Let X be a real smooth projective plane curve of degree n whose set of real points XR -I 0, and let YI0'2 - Y20'1 + "( be a determinantal representation of X that has sign E and that corresponds, as in Theorem 2.1, to a point' in J(X). Let A = (At, A2) be a non-real affine point on X. We identify the space H in the colligation with e, so that the operators AI, A2 in H are just multiplications by AI. A2, and the mapping ~ from H to E is multiplication by a vector rP in en. We want to construct a regular commutative colligation (4.1)

Vinnikov

356

The colligation conditions (3.1) and the regularity conditions (3.4) are A20"1 + 'Y)tP = 0 = tP*O"ktP (k = 1,2)

(AI0"2 -

2~Uk

"Y = 'Y + i(0"ItPtP*0"2 - 0"2tPtP·0"1) (AI0"2 - A20"1 + "Y)tP = 0 Let v

E

coker (AI 0"2 -

A20"1

+ 'Y).

(4.2) (4.3) (4.4) (4.5)

It is easily seen that

(4.6) Therefore we can normalize v so that tP = v· satisfies (4.3) if and only if

~AI = ~A2 > 0 VO"IV·

V0"2V·

(4.7)

In this case we define "Y by (4.4), and (4.5) follows. The one-point colligation (4.1) has thus been constructed. Note that we get at the output a new determinantal representation YI0"2 - Y20"1 + "Y of X. It is a fact of fundamental importance that the positivity condition (4.7) can be expressed analytically. Theorem 4.1 The condition (4.7) is satisfied if and only if fB[~~~f~(;J:) > O. In this case the new determinantal representation YI0"2 - Y20"1 + "Y defined by (4.£)-(4.4) has sign f and correspon,u to the point ( = (+ A - X in J(X). In the expressions like ( + A - X we identify the point A on X with its image in J(X) under the embedding of the curve in its Jacobian variety given by the Abel-Jacobi map p.. 8[(](w) is the theta function with characteristic (; it is an entire function on c g associated to every point ( in J(X). 8[(](w) differs by an exponential factorfrom 8«( +w). Therefore 8[(](0) =1= 0 always by Theorem 2.1; on the other hand, if the positivity condition of Theorem 4.1 is satisfied, 8«() = 8«( + A - X) =1= 0, again in accordance with Theorem 2.1. Finally, E(x,y) is the prime form on X: it is a multiplicative differential on X of order -~, ~ in x, y, whose main property is that E(x, y) = 0 if and only if x = y. See [2] or [13] for all these. Note that each factor in the expression B;~[~f~(;'!:) is multi-valued, depending on the choice of lifting from J(X) to C g , but the expression itself turns out to be well-defined. In the special case when, say, 0"2 > 0, XR divides X into two components X+ and X_ interchanged by the complex conjugation and whose affine points y = (Yl> Y2) satisfy ~Y2 > 0 and ~Y2 < 0 respectively. The "weight" B~ 0 ;;-;,.; is positive on X+ and negative on X_, the sign f = 1 and the positivity condition of T eorem 4.1 becomes A E X+ or ~A2 > 0 (see [11]). As an example, let X be the real smooth cubic (2.1). Let YI0"2 - Y20"1 + 'Y be equivalent to the representation (2.3) of sign f corresponding to the point v in the period parallelogram (v ¢. 0, v + lJ == 0), and let ~he point A on X correspond to the point

357

Vmnikov

a in the period parallelogram under the parametrization (2.2). The region for a where the positivity condition of Theorem 4.1 is satisfied depends on the component of v (see Fig. 2.1). H v E To, the admissible region is always a half of the period parallelogram; if v E Til the admissible region consists of two bands (or rather annuli) whose width depends on v; see Fig. 4.1-4.2 where the complementary admissible regions are depicted for f = 1, f = -1. H a is in the admissible region, the representation YlU2 - Y2Ul + l' is equivalent to the representation of the form (2.3) corresponding to the point v == v + a - a in the period parallelogram. T

T

-v/2+

f=1

f=1

T

f= -1

T/ 2

T/2 f

f=1

-v/2 + T/ 2

=-1

f= -1

1 Figure 4.1 : Admissible regions, v E To

1 Figure 4.2: Admissible regions, v E Tl

Using the coupling procedure we can solve now the inverse problem for regular commutative colligations with a finite-dimensional space H. Let X be a real smooth projective plane curve of degree n whose set of real points XR i: 0, and let YlU2 - Y2Ul + 'Y be a determinant~ reJJresentation of X that has sign f and corresponds to a point (in J(X). Let .x(i) = (.x~·),.x2·»(i = 1, ... ,N) be a finite sequence of non-real affine points on X. Assume that

The conditions (4.8) turn out to be independent of the order of the points .x(l), ..• , .x(N). H all the points are distinct, (4.8) can be rewritten, using Fay's addition theorem [2,13], in the matrix form f

(

i9[(](.x(i) -

W» )

9[(](0)E(.x(i) , 'XUi)

>0

(4.9)

i,j=l, ... ,N

We write down the system of recursive equations:

+ 'Y),p(i) = 0, = 2~.U~i) (k = 1,2), 'Y(i+l) = 'Y(i) + i(Ul,p(i),p(i)*U2 - U2,p(i),p(i)*ud, (.x~i)U2

-

.x~i)Ul

,p(i)*Uk,p(i)

'Y(l)

=

'Y

(4.10)

for i = 1, ... , N. It follows from Theorem 4.1 that this system is solvable (uniquely up to multiplication of ,p(i) by scalars of absolute value 1) and for each i YlU2 - Y2Ul + 'Y(i) is a determinantal representation of X that. has sign f and corresponds to the point

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358

(+ E~:~(..\(j) - ..\(j» in J(X). For each i C(i) = (..\~i),..\~i),e,4>(i),en'0"1'0"2'1'(i)'1'(i+l» is a one-point (as in (4.1» regular commutative colligation, and we can couple them by Theorem 3.1.

Theorem 4.2 Let ..\ (i)( i = 1, ... , N) be a finite sequence of non-real affine points on X satisfying (4.8), and let 1'(i), 4>(i) be determined from (4.10). Then (4.11)

is a regular commutative colligation, where

o o

..\ll) A/(2).0"/(1)

~4>(N).0"/(1) 'P = ( 4>(1)

7=

~

~lN)

i4>(N).0"/(2)

) (k=I,2),

4>(N) ) ,

1'(N+1)

(4.12)

The joint spectrum of All A2 is p(i)}~v and the output determinantal representation Vl0"2 - V20"1 + 7 of X has sign f and corresponds to the point ( in J(X), where N

(= (+ ~)..\(i)

_ ..\(i»

(4.13)

i=l

We call the solution (4.11) of the inverse problem the triangular model with discriminant curve X, input determinantal representation VI0"2-Y20"1 +1' and spectral data ..\(i)(i = 1, ... , N). The reordering of the points ..\(1), •.• , ..\(N) leads to a unitary equivalent triangular model. Furthermore, the triangular model is the unique solution of the inverse problem.

Theorem 4.3 Let C = (AI' A 2, H, 'P, en, 0"1, 0"2,1',7) be a regular commutative colligation with dimH < 00 and with smooth discriminant curve X that has real points. Let ..\(i)(i = 1, ... , N) be the points of the joint spectrum of All A2 (restricted to the principal subspace II ofC in H). Then ..\(i) are non-real affine points of X satisfying (4.8) and C is unitarily equivalent (on the principal subspace II) to the triangular model with discriminant curve X, input determinantal representation VI0"2 - Y20"1 + l' and spectral data ..\ (i)( i = 1, ... , N).

In the special case when one of the operators AI, A2 is dissipative, say 0"2 >0, the conditions (4.8) reduce to ~..\~i) > O(i = 1, ... , N) (see the comments following Theorem 4.1); Theorems 4.2-4.3 have been obtained in this case by Livsic [11]. The proof of Theorem 4.3 is based on the existence of a chain H = Ho :J HI :J ..• :J H N - l :J HN = oof common invariant subspacesof Al,A2 such that dim(Hi _ 19Hi ) = l(i = 1, ... , N) (simulataneous reduction to a triangular form; we assume for simplicity H = II). Projecting the colligation C onto the subspaces Hi-l 9Hi , we represent C as the coupling of N one-point (as in (4.1» oolligations, which forces it to be unitary equivalent to the triangular model.

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The conditions ( 4.8) determine all possible input determinantal representations (if any) of a regular commutative colligation with the discriminant curve X and operators AI, A2 having the joint spectrum A(1), ... , A(N) (on the principal subspace of the colligation). For example, let X be the real smooth cubic (2.1), and let A(I), ... ,A(N) be non-real affine points on X corresponding to the points a(I), ... ,a(N) in the period parallelogram under the parametrization (2.2). Assume m among those points lie in the upper half of the period parallelogram: ~T < ~a(i) < ~T, and k lie in the lower half: 0 < ~a(i) < ~T (m + k = N). Let YIU2 - Y2UI + '"Y be the input determinantal representations of a regular commutative colligation with the discriminant curve X and operators AI, A2 having the joint spectrum A(I), ... ,A(N). YIU2 - Y2UI + '"Y is equivalent to the representation (2.3) of sign € corresponding to the point v in the period parallelogram (v ¢. 0, v + V == 0). We may take v E To (arbitrary) if and only if k = 0 (€ = 1), or m = 0 (f = -1) (see Fig. 4.1). We may take v E TI if and only if - ~a

(1)

- ... - ~a

(N)

~v + 2m2+ k 0.0 (z=O, ... ,N-l),

00

E(A(i) - A(i» converges ,6«( + E(A(i) - A(i»):F 0, i=l i=l 6

(

t

N

(+ ~(A(i) -

A(i»

+ Ei 10

,=1

0

: ( ~) ~

~

)

dS:F 0 (t E [0, I])

(5.10)

~

where W,Wb ... ,wg are as before (if N < 00 the second condition is not needed). Write down the system of recursive equations (5.2) followed by the system of differential equations (5.5)

+ "()t/J(i) = 0, t/J(i)·Ukt/J(i) = 2S3'ALi) (k = 1,2), "(i+l) = "(i) + i( U1 t/J(i)t/J(i)* U2 - U2t/J(i)t/J(i)* U1), (Aii)U2 - A~i)Ul

= "(, i = 1, ... ,Nj

"(1)

(C1(t)U2 - C2(t)U1 + ,,()t/J(t) = 0, * dy,.(c(t» t/J(t) Ukt/J(t) = E w(c(t» (k = 1,2),

~; = "(0)

(if N


a(I~112 + 1~212)1/2} for some a> 0, and R(t~ht~2,tZ) = t-1R(~h~2'Z) for all t E C,t '" o.

366

Vinnikov

2) For any affine point Y = (YbY2) on X, S(6,6,6Yl +6Y2) maps L(y) = coker (YI0"2 Y20"1 +'Y) into L(y) = coker(YI0"2-Y20"1 +1') and the restriction S(6,6,6Yl +6Y2) I L(y) is independent of {b 6 (6, {2, 6Yl + {2Y2) E Ka}. 3) For any {b6 E R, S(6,6,z) is a meromorphic function of z on the complement of the real axis and

S(6, 6, Z)({IO"I S(6, 6, Z)(60"1 ({b {2, 6Yl

+ 60"2)S(6, 6, zr :5 60"1 + {20"2 + 60"2)S(6, 6, z)* = 60"1 + 60"2

(S 0), (S 0 this has been obtained by Livsic [12]. Using (6.3)-(6.4) we see that Theorem 5.4 on the reduction to the triangular model is equivalent to the following: for every matrix function S( 6, 6, z) satisfying the conditions of Theorem 6.3, there exists a spectral data ..\(i)(i = 1, ... ,N; N :5 oo),c(t) = (Cl(t),~(t»(O:5 t :5 I) satisfying (5.10), such that 'Y(l) =.:y and

S({b6,z) =

n I+i({10"1+{20"2) N

(

.=1

loo exp a

X

(.

Z(60"1

fjJ(i)fjJ(i)* (i) (i)

6..\1

+ {2..\2

)

- z

fjJ(t)fjJ(t)*) z dt

+ 60"2) {lCl () t + {2C2 () t -

(6.8)

where 'Y(i),fjJ(i),'Y(t),fjJ(t) are determined by (5.11). Now, functions of several complex variables do not admit a good factorization theory. However, w~ see from Theorem 6.4 that the complete characteristic function reduces to the function on the one-dimensional complex manifold X. We shall therefore reduce (6.8) to the factorization theorem on a real lliemann surface. We first want to express the contractivity and isometricity properties (6.6) of the complete characteristic function in terms of the joint characteristic function. To this

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367

end we introduce a hermitian pairing between the fibers L(yCl», L(yC2» of the line bundle L(y) = coker(Yl0'2 - Y20'1 + 'Y) over non-conjugate affine points yCl) = (YP),y~1»,yC2) = (Y12),y~2» on X:

(6.9) This is in fact independent (see (4.6» of el, e2 E R. In particular, taking y = yCl) = y(2) , we get an (indefinite) scalar product on the fiber L(y) over non-real affine points yon X. We also introduce a hermitian pairing between the fibers L(y), L(y) over conjugate affine points:

(6.10) This is again independent of 6,6 E R, and we get in particular a scalar product on the fiber L(y) over real affine points y on X (to get a value in (6.10) we have to choose, of course, a local parameter on X at the point y = (Yb Y2».

Theorem 6.5 Let S(eb e2, z) be a matrix function satisfying the conditions 1)-2) of Theorem 6.9, and let the function S(y) be defined by (6.7). Then S(6,e2,Z) satisfies (6.6) if and only if S(y) satisfies the following: for all affine points y, yCl), ... , yeN) on X in its region of analyticity (yCi) =f. yb) ) ([uCi)S(yCi», u b )S(yb»l~;) I/(J»). . ,

t,J=l, ... ,N

:5 ([u Ci ), u(j)l~;) y{i») t,J=l, .. ... ,N 1

(u Ci ) E L(yCi»; i = 1, ... , N),

[uS(y),vS(Y)l~'ii = [u,vl~,ii (u E L(y),v E L(y»

(6.11)

Let now S(y) be the joint characteristic function of a regular commutative colligation with discriminant curve X, input determinantal representation Yl0'2 - Y20'1 + 'Y and output determinantal representation YI0'2 - 1;20'1 +;y, where the representations YI0'2 Y20'1 +'Y, YI0'2 -Y20'1 +;y have sign e (the input and the output determinantal representation have always the same sign) and cOE"espond, as in Theorem 2.1, to the points (,( in J(X) (8( () =f. 0, 8( () =f. 0, ( + "( = e, ( + ( = e). Since ( and ( are, up to a constant translation, the images of the line bundles Land t in the Jacobian variety under the Abel-Jacobi map 1', and S is a mapping of L to t, it follows that S can be identified, up to a constant factor of absolute value 1, with a (scalar) multivalued multiplicative function s(x) on X, with multipliers of absolute value 1 corresponding to the point ( - ( in J(X). More precisely, let A b ..• , A g, B}, ... , Bg be the chosen canonical integral homology basis on X, let Z be the 9 X 9 period matrix of J(X) (the period lattice A C C g is spanned by the 9 vectors of the standard basis and the 9 columns of Z), and let ( = b + Z a, ( = b+ Z ii, where a, b, ii, b are vectors in Rg with entries ai,a;,bi,bi respectively; then the multipliers x. of s(x) over the basis cycle are given by

X.(Ai) = exp( -211"i(ai - ai» (i = 1, ... , g), X.(Bi ) = exp(211"i(bi - bi» (i = 1, ... ,g)

(6.12)

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368

See e.g. [2] for more details. We call s( x) the normalized joint characteristic function of the colligation (since it arises from the joint characteristic function by choosing sections of L and L with certain normalized zeroes and poles). We have essentially seen in Theorem 4.1 that the pairing (6.9) on the line bundle can be expressed analyticallYj the same is true of the pairing (6.10). We obtain thus from Theorem 6.5 a complete analytic description of normalized joint characteristic functions of regular commutative colligations. Theorem 6.6 Let YIU2 - Y2UI + 'Y be a determinantal representation of X that has sign E and corresponds to the point ( in J(X), and let ( be another point in J(X), 9«() =/: 0, (+'( = e. A multivalued multiplicative function s(x) o.n X with multipliers of absolute value 1 corresponding to the point ( - ( is the normalized joint characteristic function of a regular commutative colligation with discriminant curve X, input determinantal representation YIU2 - Y2UI + 'Y and an output determinantal representation YIU2 - Y2UI + i that has sign E and corresponds to the point ( in J(X) if and only if: 1) s(x) is holomorphic outside a compact subset of affine points of X. 2) s(x) is meromorphic on X\XR, and for all points x, X(I), ... , x(N) on X in its region of analyticity (x(i) =/: xb») E

(S(x(i»S(Xb»

s(x)s(~) =

i~[(](x(i) - ;vi) ) < 9[(](0)E(x(i),xb» i,j=I ..... N -

E (

i9[(](x(i) - ;vi) ) 9[(](0)E(x(i),xb»

, i,j=I •...•N

1

(6.13)

In the special case when one of the operators AI, A2 in the colligation is dissipative, say U2 > 0, the "weights" 8 (~)~(:3!)' 8(~ (O)~(:'3!) are positive on X+ and negative on X_ (see comments following Theorem 4.1), and it turns out that the matrix condition in (6.13) can be replaced by (6.14) Els(x)1 ;:; E (x E X+) We conjecture that in general the matrix conditIon is equivalent to

t

i9[(](x - x)

E9[(](0)E(x,~)

< -

i9[(](x - x) E 9[(](0)E(x,x)

(6.15)

(x E X\XR)

Let now X be a compact real Riemann surface (i.e. a compact Riemann surface with an antiholomorphic involution x t-+ Xj for example, a real smooth projective plane curve). Let XR be the set of fixed points of the involutionj assume XR =/: 0. Let (, ( be two points in J(X), 9«() =/: 0,9«() =/: 0,( +"( = e,( + '( = e (the half-period e of Theorem 2.1 is defined for every real Riemann surface). A multivalued multiplicative function s( x) on X with multipliers of absolute value 1 corresponding to the point ( - ( is called semicontractive, or, specifically, «(, ()-contractive, if it is meromorphic on X\XR, and for all points x, X(I), ••• , x(N) on X\XR (x(i) =/: x(j):

(

(') -(-') i9[(](x(i) - ;vi) ) s(x • )s(x J ) 9[C](0)E(x(i), x(j)

s(x)s(~)

= 1

( i9[(](x(i) - ;m) ) i,j=I .....N;:;

9[(](0)E(x(i), ;m')

iJ=I .....N'

(6.16)

369

Vinnikov

Theorem 6.6 states that normalized joint characteristic functions of regular commutative colligations with a smooth discriminant curve (that has real points) are precisely semicontractive functions on the discriminant curve (for sign f = 1) and their inverses (for sign f = -1). The factorization (6.8) of the complete characteristic function follows from the following factorization theorem for semicontractive functions on the real lliemann surface X. Theorem 6.7 Let s( x) be a «(, ()-contractive function on X. Then sex) =

IT

(exp (1I"im(i)t(A(i)

+ .W» + 1I"i m(i)t Hm(i») exp (-211"(A(i) _

i=1

x exp

2 (Wl(y), ... ,wg(y» d () (- 11" ~1 ni v i=O ( ») +z'1 d lnE(x, () y)dvy L...J

X.

W

y -

( )

Y

'1

2 1I"Z

xR

A(i»tyx) E(x, A(~») E(x, A('»

(Wl(y), ... ,Wg(y»y d ( ) ( ) X V Y W y

y

XR

W

(6.17)

Y

Here A(i)(i = 1, ... ,NjN:5 00) are the zeroes ofs(x) onX\XR and v is a uniquely determined finite positive Borel measure on XR; WI, . , . ,Wg are the chosen basis for holomorphic differentials on X; W is a real differential on X, defined, analytic and non-zero in a neighbourhood of supp v C XR, whose signs on different connected components X o, . .. , X k _ 1 of XR correspond to the real torus in J(X) to which the points (, ( belong [tB); Z = !H +iy-l (H, Y real) is the g x g period matrix of J(X); m(i)( i = 1, ... ,N), ni(i = 0, ... ,k - 1) are integral vectors depending on the choice of lifting of the points A(i) and the components Xi respectively from J(X) = c g/ A to C g. Furthermore, the following hold: i8[( + E~=I(A(j) - ,X(i»)(A(Hl) - A(i+1»

,

,

8[( + Ei=I(,X(i) - ,X(i»)(O)E(A(Hl), A(i+1» 00

> 0 (z = 0, ... , N - 1),

00

E(A(i) - A(i» converges ,8«( + E(A(i) ~ A(i»):F 0 (if N = 00), i=1 i=1 8

(

A(i»

.=1

i=1

+i

B

N

( = (+ E(A(i) -

1 ( ~) : W(II)

N

(+ ~:::CA(i) -

A(i»

+i

~

)

dV(Y):F 0 (for all Borel sets B C XR),

1 w(y~) : ~

XR

~ ~

dv(y)

(6.18)

When X is a real smooth projective plane curve (and sex) is holomorphic outside a compact subset of affine points of X), the two factors in (6.16) are the normalized joint characteristic functions ofthe colligations (5.4) and (5.7) respectively (c : [0, I) XR is the left-continuous non-decreasing function determined by v(B) = m(c- 1 (B» for Borel sets B C XR, where m is the Lebesgue measure on [O,lj, I = V(XR». Decomposing the measure v into singular and absolutely continuous parts (with respect to the measures induced on XR by the usual Lebesgue measure through local coordinates), we obtain the

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370

factorization of a semi contractive function into a Blaschke product, a singular inner function and an outer function, generalizing the lliesz-Nevanlinna factorization for bounded analytic functions in the unit disk (see e.g. [7]). Our factorization is better compared though to Potapov factorization for J-contractive matrix functions (see [15]), since the i9[,](z-x) t . al ·t· t· h I the wet.ghts 9,i9,0 Ez-x z,x ' 9['](O)E(z,z) are no , 10 gener , POSl lve or nega lve everyw ere. n special case of (6.14), the Blaschke product - singular inner facter - outer factor decomposition was known ([22,5,6]), without, however, explicit formulas for the factors in terms of the prime form E(x, y). It is my pleasure to thank Prof. M.S.LivJic for many deep and interesting discussions.

References [1] Brodskii,M.S., Livsic,M.S.: Spectral analysis of nonselfadjoint operators and intermediate systems, AMS Transl. (2) 13, 265-346 (1960). [2] Fay,J.D.: Theta Functions on Riemann Surfaces, Springer-Verlag, Heidelberg (1973). [3] Griffiths,P., Harris,J.: Principles of Algebraic Geometry, Wiley, New York (1978). [4] Harte,R.E.: Spectral mapping theorems, Proc. Roy. Irish Acad. (A) 72, 89-107 (1972).

[5] Hasumi,M. : Invariant subspace theorems for finite lliemann surfaces, Canad. J. Math. 18, 240-255 (1986).

[6] Hasumi,M. : Hardy Classes on Infinitely Connected Riemann Surfaces, SpringerVerlag, Heidelberg (1983).

[7] Hoffman,K.: Banach Spaces of Analytic Functions, Prentice Hall, Englewood Cliffs, NJ (1962). [8] Kravitsky,N. : Regular colligations for several commuting operators in Banach space, Int. Eq. Oper. Th. 6, 224-249 (1983). [9] Kravitsky,N.: On commuting integral operators, Topics in Operator Theory, Systems and Networks (Dym,H., Gohberg,l., Eds.), Birkhauser, Boston (1984).

[10] Livsic,M.S., Jancevich,A.A.: Theory of Operator Colligations in Hilbert Space, Wiley, New York (1979). [11] Livsic,M.S.: Cayley-Hamilton theorem, vector bundles and divisors of commuting operators,Int. Eq. Oper. Th. 6, 250-273 (1983).

[12] LivSic,M.S.: Commuting nonselfadjoint operators and mappings of vector bundles on algebraic curves, Operator Theory and Systems (Bart,H., Gohberg,l., Kaashoek,M.A., Eds.), Birkhiiuser, Boston (1986). [13] Mumford,D.: Tata Lectures on Theta, Birkhiiuser, Boston (Vol. 1, 1983; Vol. 2, 1984). [14] Nikolskii,N.K.: Treatise on the Shift Operator, Springer-Verlag, Heidelberg (1986).

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[15] Potapov,V.P.: The mulptiplicative structure of J-contractive matrix functions, AMS Transl. (~) 15, 131-243 (1960). [16] Taylor,J.L. : A joint spectrum for several commuting operators, J. of Funct. Anal. 6, 172-191 (1970). [17] Vinnikov ,V.: Self-adjoint determinantal representations of real irreducible cubics, Operator Theory and Systems (Bart,H., Gohberg,I., Kaashoek,M.A., Eds.), Birkhauser, Boston (1986). . [18] Vinnikov,V.: Self-adjoint determinantal represent ions of real plane curves, preprint. [19] Vinnikov,V. : Triangular models for commuting nonselfadjoint operators, in preparation. [20] Vinnikov,V. : Characteristic functions of commuting nonselfadjoint operators, in preparation. [21] Vinnikov, V. : The factorization theorem on a compact real Riemann surface, in preparation. [22] Voichick,M., Zalcman,L. : Inner and outer functions on Riemann surfaces, Proc. Amer. Math. Soc. 16, 1200-1204 (1965). [23] Waksman,L.: Harmonic analysis of multi-parameter semigroups of contractions, Commuting Nonselfadjoint Operators in Hilbert space (Livsic,M.S., Waksman,L.), Springer-Verlag, Heidelberg (1987).

DEPARTEMENT OF THEORETICAL MATHEMATICS, WEIZMANN INSTITUTE OF SCIENCE, REHOVOT 76100, ISRAEL

E-mail address:[email protected] 1980 Mathematics Subject Classification (1985 Revision). Primary 47 A45, 30D50j Secondary 14H45, 14H40, 14K20, 14K25, 30F15.

372

Operator Theory: Advances and Applications, Vol. 59 © 1992 Birkhiiuser Verlag Basel

ALL (?) ABOUT QUASINORMAL OPERATORS Pei Yuan Wu 1) Dedicated to the memory of Domingo A. Herrero (1941-1991)

A bounded linear operator T on a complex separable Hilbert space is quasinormal if T and T *T commute. In this article, we survey all (?) the known results concerning this class of operators with more emphasis on recent progresses. We will consider their various representations, spectral property, multiplicity, characterizations among weighted shifts, Toeplitz operators and composition operators, invariant subspace structure, double commutant property, commutant lifting property, similarity, quasisimilarity and compact perturbation, and end with some speculations on possible directions for further research.

1. INTRODUCTION

The class of quasi normal operators was first introduced and studied by A. Brown [4] in 1953. From the definition, it is easily seen that this class contains normal operators (TT * = T *T) and isometries (T *T = I). On the other hand, it can be shown [36, Problem 195] that any quasinormal operator is subnormal, that is, it has a normal extension. Normal operators and isometries are classical objects: Their properties have been fully explored and their structures well-understood. It has also been widely recognized that subnormality constitutes a deep and useful generalization of normality. After two-decades' intensive study by various operator theorists, the theory of subnormal ope~ators has matured to the extent that two monographs [17, 18] have appeared which are devoted to its codification. People may come to suspect whether the in-between quasinormal operators would be of any interest to merit a separate survey paper like this one. The structure of quasinormal operators is, as we shall see below, l)This research was partially supported by the National Science Council of the Republic of China.

Wu

373

indeed very simple. They are certainly not in the same league as their big brothers : Their theory is not as basic as those of normal operators and isometries and also not as deep as subnormal ones. However, we will report in subsequent discussions some recent progresses in the theory of quasinormality which serve to justify the worthwhileness of our effort. One recent result (on the similarity of two quasinormal operators) establishes a connection between the theories of quasinormal operators and nest algebras. Another one (on their quasi similarity) uses a great deal of the analytic function theory. These clearly show that there are indeed many interesting questions which can be asked about this class of operators. It used to be the case that the study of quasi normal operators was pursued as a step toward a better understanding of the subnormal ones. The recent healthy developments indicate that quasinormal operators may have an independent identity and deserve to be studied for their own sake. The interpretation of our title "ALL (?) ABOUT QUASINORMAL OPERATORS" follows the same spirit as that of Domingo Herrero's paper [39] : The "ALL" is interpreted as "all the author knows about the subject", and the question mark "?" means that we-never really know "all" about any given subject. The paper is organized as follows. We start in Section 2 with three representations of quasinormal operators. One of them is the canonical representation on which all the theory is built. Section 3 discusses the (essential) spectrum, various parts thereof, (essential) norm and multiplicity. Section 4 gives characterizations of quasinormality among several special classes of operators, namely, weighted shifts, Toeplitz operators and composition operators. Section 5 then treats various properties related to the invariant subspaces of an operator such as reflexivity, decomposability, (bi)quasitriangularity and cellular-indecomposability. The three operator algebras {T}', {T}" and Alg T of a pure quasinormal operator T are described in Section 6. Then we proceed to consider properties relating a quasi normal operator to operators in its commutant. One such property concerns their lifting to its minimal normal extension. We also consider the quasinormal extension for subnormal operators as developed by Embry-Wardrop. Sections 7 and 8 are on the similarity and quasisimilarity of two quasinormal operators. Section 9 discusses the problems when two quasinormal operators are approximately equivlaent, compact perturbations and algebraically equivalent to each other. We conclude in Section 10 with some open problems which seem to be worthy of exploring. This paper is an expanded version of the talk given in the WOTCA at Hokkaido University. We would like to thank Professor T. Ando, the organizer, for his

Wu

374

invitation to present this talk and for his efforts in organizing the conference. 2. REPRESENTATIONS We start with the canonical representation for quasi normal operators first obtained by A. Brown [4]. This representation is the foundation for all the subsequent developments of the theory. THEOREM 2.1. An operator T on Hilbert space H is quasinormal if and

only ifT is unitarily equivalent to an operator of the form

where N is normal and A is positive semidefinite. If A is chosen to be positive, then N and A are uniquely determined (up to unitary equivalence). Recall that A is positive semidefinite (resp. positive definite) if (Ax, x) ~ 0 (resp. (Ax, x) > 0) for any vector (resp. nonzero vector) x. In fact, in the preceding theorem N and A may be chosen to be the 1 restrictions of T and (T*T)7J. to their respective reducing subspaces nfh ker (Tn *Tn -

T~n*) and H e (ker T space, then

CD

rail'T). If A is the identity operator on a one-dimensional

[

~0

AO

reduces to the simple unilateral shift S. (Later on, we will also consider S as the operator of multiplication by z on the Hardy space H2 of the unit disc.) For convenience, we will denote

Wu

375

by S ® A without giving a precise meaning to the tensor product of two operators. Note that S ® A is completely nonnonnal, that is, there is no nontrivial reducing subspace on which it is normal. We will call the uniquely determined N and S ® A the nonnal and pure parts of T, respectively. If T is an isometry, then these two parts coincide with the unitary operator and the unilateral shift in its Wold decomposition. In terms of this representation, it is easily seen that every quasinormal operator N $ (S ® A) is subnormal with minimal normal extension

N$

AO AO AO

where a box is drawn around the (0, 0) -entry of the matrix. Since

is the (unique) polar decomposition of S ® A (with the two factors having equal kernels), an easy argument yields the following characterization of quasi normality [36, Problem 137]. THEOREM 2.2. An operator with polar decomposition UP is quasinonnal

if and only ifU and P commute. There are other representations for quasi normal operators. Since every positive operator can be expressed as the direct sum of cyclic positive operators, this implies that every pure quasinormal operator is the direct sum of operators of the form S ® A, where A is cyclic and positive definite. (Recall that an operator T on H is cyclic

Wu

376

if there is a vector x in H such that V{Tnx : n ~ O} = H.) The second representation which we now present will be for this latter type of operators. By the spectral theorem, any cyclic positive definite operator A is unitarily equivalent to the operator of multiplication by t on L2(,,), where" is some positive Borel measure on an interval [0, a) in IR with" ({O}) = O. Let II be the measure on ( defined by dv{z) = dOd,,(t), where z = teiO, and let K = V {Izlmzn : m, n ~ O}

J

in L2(11). Then, obviously, K is an invariant subspace for M, the operator of multiplication by z on L2(11). Finally, let T A = M IK. THEOREM 2.3. For any cyclic positive definite operator A, T A is a pure quasinormal operator. Conversely, any pure quasinormal operator S ® A with A cyclic is unitarily equivalent to T A'

This representation is obtained in [19, Theorem 2.4). The appearance of the space K above is not too obtrusive if we compare it with the space in the statement of Proposition 3.3 below. We conclude this section with the third representation. It applies to pure quasinormal operators S ® A with A invertible. This is originally due to G. Keough and first appeared in [19, Theorem 2.8). Let A be a positive invertible operator on H, and let H be the class of

1

sequences {xn}~=O with xn in H satisfying n!0llAnXnll2
i and AD ij = DH 1 j+ 1A for i ~ j.

As for {T}" and Alg T, their characterizations lie deeper. Recall that the simple unilateral shift S satisfies {S}' = {S}" = Alg S = {t/I{S) : t/I E HCO } (cf. [36, Problems 147 and 148]). That the commutant and double commutant cannot equal for general (higher-multiplicity) unilateral shifts is obvious. The next theorem says that the remaining equalities (with a slight modification) still hold for any pure quasinormal operator. THEOREM 6.2. For any pure quasinormal operator T = seA, the equalities {T}" = Alg T = {t/I{T) : t/I E ~} hold, where r = IITII and H~ denotes the Banach algebra o/bounded analytic junctions on {z E ( : Izl < r}. Thus, in particular, operators in {T}" = Aig T are of the form

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383

aOI alA

0

aOI

0

~A2 alA a O!

where the an's are the Fourier coefficients of a function cP(z) = l:n:O anzn in H~. These results were proved in [19]. For nonpure quasi normal operators, the double commmant property ({T}" = Alg T) does not hold in general. Actually, this is already the case for normal operators; the bilateral shift U on L2 of the unit circle is such that {U}" = {1/J(U) : 1/J E Loo} and Alg U = {cP(U) : cP E Hoo}. A complete characterization of quasi normal operators satisfying the double commutant property is given in [19, Theorem 4.10]. The conditions are too technical to be repeated here. We content ourselves with the following special case which was proved earlier in [52]. PROPOSITION 6.3. Any nonunitary isometry has the double commutant

property. We next consider the commutant lifting problem: If T is a quasinormal operator on H with minimal normal extension N, when is an operator in {T}' the restriction to H of some operator in {N}'? That this is not always the case can be seen from the following example. Let T = S ® A, where A = [~ ~], and X = diag (B, ABA-I, A2BA-2, ... ), where B =

[~

n.

Since AnBA-n =

[~ (I/i)n]

for n

~ 0, X is indeed a bounded

operator. That X belongs to {T}' follows from Proposition 6.1. A simple computation shows that if X can be lifted to an operator Y in the commutant of the minimal normal extension

of T, then Y must be of the form diag ( ..• , A-2BA2, A-I BA , B, ABA-1, A2BA-2 ..• ). However, as

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for n ~ 0, this operator cannot be bounded. This shows that X cannot be lifted to {N}'. Note that in this example T is even a pure quasinormal operator with multiplicity 2. A complete characterization of operators in {T}' which can be lifted to {N}' is obtained by Yoshino [62, Theorem 4]. THEOREM 6.4. Let T be a quasinormal operator with minimal normal extension N and polar decomposition T = UP. Then X E {T}' can be lifted to Y E {N}' if and only if X commutes with U and P. Moreover, if this is the case,then Y is unique

and IIYII

= IIXII·

In particular, if T is an isometry, then operators in {T}' can always be lifted [27, Corollary 5.1]. These results are subsumed under Bram's characterization of commutant lifting for subnormal operators [3, Theorem 7]. Another version of the lifting problem asks whether two commuting quasinormal operators have commuting (not necessarily minimal) normal extensions. An example of Lubin [43] provides a negative answer. Indeed, the two quasinormal operators T I and T 2 he constructed are such that both are unitarily equivalent to S e 0, where 0 denotes the zero operator on an infinite-dimensional space, TIT 2 = T 2T I = 0 and T 1

+

T 2 is not hyponormal. Again, a complete characterization in terms of the

polar decomposition is given in [62, Theorem 5]. THEOREM 6.5. Let T I and T2 be commuting quasinormal operators with

polar decompositions TI

= UIP I

and T2

= U2P2'

Then TI and T2 have commuting

normal extensions if and only ifU I and PI both commute with U2 and P 2' In this connection, we digress to discuss another topic which may shed some light on the commutant lifting problem. As is well-known, every subnormal operator has a unique minimal normal extension [36, Problem 197]. That it also has a unique minimal quasi normal extension seems to be not so widely known. This fact is due to Embry-Wardrop [28, Theorems 2 and 3]. THEOREM 6.6. Let T be a subnormal operator with minimal normal extension N on H. IfK = V {!:J=o(N*N)jXj : Xj E H, n ~ OJ, then NIK is a minimal

quasinormal extension of T and any minimal quasinormal extension of T is unitarily equivalent to N IK. Moreover, N is also the minimal normal extension ofN IK. Thus, in particular, the lifting of the commutant for subnormal operators

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can be accomplished in two stages: first lifting to the commutant of the minimal quasi normal extension and then the minimal normal extension. Studies of other properties of subnormal operators along this line seem promising but lacking. A problem which might be of interest is to determine which subnormal operator has a pure quasi normal extension. As observed by Conway and Wogen [58, p.169], subnormal unilateral weighted shifts do have this property. We conclude this section with properties of a class of operators considered by Williams [57, Section 3]. A result which is of interest and not too difficult to prove is the following. THEOREM 6.7. 1fT is a quasinormal operator, N is normal and TN = NT, then T + N is subnormal. Starting from this, he went on to consider operators of the form T + N, where T is pure quasi normal and N is a normal operator commuting with T. It turns out that such operators have a fairly simple structure. If we express TasS ® A on H 49 H 49 ••• and use Proposition 6.1, we can show that N must be of the form NO 49 NO 49 . . .. An easy consequence of this is THEOREM 6.8. 1fT is a pure quasinormal operator, N f 0 is normal and TN = NT, then T + N is not quasinormal. For other properties of such operators, the reader is referred to [57]. 7. SIMILARITY In this section and the next two, we will consider how two quasi normal operators are related through similarity, quasisimilarity and compact perturbation. We start with similarity. For over a decade, the problem whether two similar quasi normal operators are actually unitarily equivalent remains open [41]. This is recently solved in the negative in [12]. In fact, a complete characterization is given for the similarity of two quasi normal operators. Note that the similarity of two normal operators or two isometries implies their unitary equivalence (even the weaker quasisimilarity will do). For normal operators, this is a consequence of the Fuglede-Putnam theorem [36, Corollary to Problem 192]; the case for isometries is proved in [40, Theorem 3.1]. On the other hand, there are similar subnormal operators which are not unitarily equivalent [36, Problem 199]. Against this background, the result on quasinormaI operators should have more than a passing interest.

386

Wu

THEOREM 7.1. For j = 1, 2, let T j = Nj

G)

(S

®

Aj) be a quasinormal

operator, where Nj is normal and Aj is positive definite. Then TI is similar to T2 ifand only ifNI is unitarily equivalent to N2, q(A I ) = q(A2 ) and dim ker (AI - AI) = dim ker

(A2 - AI) for any A in q(A I ). Thus, in particular, similarity of quasi normal operators ignores the multiplicity of the operator Aj in the pure part except those of its eigenvalues. From this observation, examples of similar but not unitarily equivalent quasi normal operators can be easily constructed. One such pair is T 1 = S ® A and T2 = S ® (A G) A), where A is the operator of multiplication by t on L2[0,1). As for the proof, we may first reduce our consideration to pure quasinormal operators by a result of Conway [16, Proposition 2.6): Two subnormal operators are similar if and only if their normal parts are unitarily equivalent and their pure parts are similar. For the pure ones, the proof depends on a deep theorem in the nest algebra theory. Here is how it goes. Recall that a collection K of (closed) subspace of a fixed Hilbert space H is a nest if (1) {O} and H belong to )I, (2) any two subspaces M and N in K are comparable, that is, either M ~ N or N ~ M, and (3) the span and intersection of any family of subspaces in K are still in No For any nest )I, there is associated a weakly closed algebra, Alg )I, consisting of all operators leaving invariant every subspace in K; Alg K is called the nest algebra of No The study of nest algebra is initiated by J.R.Ringrose in the I960s. Since then, it has attracted many researchers. A certain maturity is finally reached in recent years. The monograph [23) has a comprehensive coverage of the subject. Before stating the Similarity Theorem which we are going to invoke, we need some more terminology of the theory. A nest K is continuous if every element N in K equals its immediate predecessor N :: V {N' E JI: N' ¥N}. Two nests K and Jf on spaces HI and H2 are similar if there is an invertible operator X from HI onto H2 such that XX

= Jf.

A major breakthrough in the development of the theory is the proof by Larson [42) that any two continuous nests are similar. This is generalized later by Davidson [22) to the Similarity of any two nests: K and Jf are similar if and· only if there is an order-preserving isomorphism (J from K onto Jl such that for any subspaces N1 and N2 in Kwith NI ~ N2 the dimensions of N2 e Nl and fJ(N 2) e fJ(N 1) are equal. In particular, this says that the similarity of nests depends on the order and the dimensions (of the

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atoms) of 'the involved nests but not on their multiplicity, (A multiplicity theory of nests can be developed via the abelian von Neumann algebra generated by the orthogonal projections onto the subspaces in the nest.) This may explain why the Similarity Theorem has some bearing on our result. Its proof is quite intricate. Before embarking on the proof of our result, we need a link relating pure quasi normal operators to nest algebras so that the Similarity Theorem can be applied. For any positive definite operator A on H, there is associated a natural nest JlA , the one generated by all subspaces of the form EA([O,tj)H, t

~ 0,

where E A(·) denotes the

spectral measure of A. The result we need is due to Deddens [25]. It says that the nest algebra Alg JIA consists exactly of operators T satisfying sUPn~O IIA ~ A-nil < (I). Now we are ready to sketch the proof of Theorem 7.1. If Al and A2 are positive definite operators on HI and H2 satisfying O'(AI) = 0'(A 2) and dim ker (AI - AI) = dim ker (A 2 - AI) for A in O'(A I ), then define the order-preserving isomorphism () from JIA to JIA by I 2 () (E A [O,A]H 1) = E A [O,A]H 2 I 2

if A E 0'(1\1)

and () (E A [O,A )H 1) = E A [O,A )H2 if A is an eigenvlaue of AI' I 2 Our assumption guarantees that () is dimension-preserving. Thus it is implemented by an invertible operator X by the Similarity Theorem. Letting [ 0 X-I] ' andY=XO

we have Y

E

Alg JlA . Therefore, Deddens' result implies that sUPn~O IIAnYA-nll
1. This is established through modifying

the proof of a result of Sz.-Nagy and Foias [51] that IlS(n) -< S for any Il, IIII > 1, and n, 1 ~ n ~ 00. On the other hand, following our assumptions, (b) is the same as S ® D2 -< (S

®

C2) E9 (S

®

D2). The proof, based on the fact that IIC211 ~ m(D 2), is easier (d. [12,

Lemma 3.14 (a)]). We next turn to the quasisimilarity of general quasi normal operators. The following theorem gives a complete characterization. THEOREM 8.2. For j = 1, 2, let T j = Nj E9 (S ® Aj) be a quasinormal

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390

operator. Let a = min{m(A I ), m(A 2)} and d

= max{dim ker (AI -

aI), dim ker (A 2 -

aI)}. Then T I is quasisimilar to T 2 if and only if N 1 is unitarily equivalent to N2' S

GD

Al is densely similar to S GD A2 and one of the foUowing holds: (1) S GD Al is quasisimilar to S GD A2; (2) d = 0 and o-(N I ) has a limit point in the disc

{z E ( : Izl

~

a};

(3) d > 0 and the absolutely contin uow unitary part of NI / a does not

vanish; (4) d > 0 and the completely nonunitary part ofNI/a is not of class CO. Some explanations for the terminology used above are in order. normal operator M on H can be decomposed as M

Any

= MI $ M2 $ M3, where M I , M2 and

M3 act on EM(Il)H, E M ( lJI)H and EM((\D)H, respectively (II is the open unit disc on the plane).

M 1, being a completely nonunitary contraction, is called the completely

nonunitary part of M. The unitary M2 can be further decomposed as the direct sum of an absolutely continuous unitary operator and a singular unitary operator. These are the parts referred to in conditions (3) and (4) in the above theorem. nonunitary contraction T is of class Co if 4>(T)

= 0 for some 4> E JfiO.

A completely

(For properties of

such operators, the reader is referred to [50].) The proof of the sufficiency of the conditions in Theorem 8.2 involves a great deal of function-theoretic arguments. For simplicity, we will present one typical example for each of the conditions (2), (3) and (4) followed by a one-sentence sketch of its proof which somehow gives the general flavor of the arguments. EXAMPLE 8.3. IfN is the diagonal operator diag(d n ) on

r, where {dn }

is a sequence satisfying 0 < Id n I ~ c < I for aU n and converging to 0, then S $ N -< N. The operator X : H2 $

r . . r defined by

X(f$ {an}) = {cn(f(dn )

+ an exp(-l/Idnl))}

can be shown to be injective, with dense range and satisfying X(S .$ N) = NX. EXAMPLE 8.4. If N is the operator of multiplication by e it on L2(E), where E is a Borel subset of the unit circle, then S $ N -< N. The operator X : H2 $ L2(E) ... L2(E) required is defined by

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391

X(f ED g) = (fl E)

+ tPg,

J

where ¢J is a function in Loo(E) such that ¢J 1= 0 a.e. on E and E log I¢J I = EXAMPLE 8.5. If N is the diagonal operator diag( dn ) on

00.

r, where

{dn }

is a sequence of points in the open unit disc accumulating only at the unit circle and satisfuing En (1-1 dn I) = 00, then S ED N -< N. -+

The proof for this case is the most difficult one. The operator X : H2 defined by

r

ED

r

1

X(f ED {an}) = {f(dn )(I-1 dn 12)~/n

+ anb nexp(-I/(I-ldn 1)2)},

where {bn} is a bounded sequence of positive numbers satisfying lim sUPn IB( dn ) I/nbn ~ 1 for any Blaschke product B (the existence of {bn} is proved in [12, Lemma 4.8]), will meet all the requirements. The difficulty lies in showing the injectivity of X. 9. COMPACT PERTURBATION

a

Two operators T 1 and T 2 are approximately equivalent (donoted by T 1 ~

T 2) if there is a sequence of unitary operators {Un} such that II Un*T 1Un - T211-+ OJ they

a

are approximately similar (donoted by T 1 ~ T 2) if there are invertible operators Xn such that sup {"Xn",

IIX~II1}