Table Of ContentOperator Theory: Advances and Applications 47
Editor
I. Gohberg
Extension and Interpolation
of Linear Operators and
Matrix Functions
Extension and
Interpolation of
Linear Operators and
Matrix Functions
Edited by
I. Gobberg
1990 Springer Basel AG
Editor's address:
Prof. 1. Gohberg
Raymond and Beverly Sackler Faculty of Exact Sciences
School of Mathematical Sciences
Tel Aviv University
69978 Tel Aviv, Israel
CIP-litelaufnabme der Deotschen Bib6othek
Extension and Interpolation of Linear Operators and Matrix Functions 1
ed. by 1. Gohberg.
(Operator theory ; Voi. 47)
ISBN 978-3-7643-2530-5 ISBN 978-3-0348-7701-5 (eBook)
DOI 10.1007/978-3-0348-7701-5
NE: Gohberg, Izrael' [Hrsg.]; GT
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© 1990 Springer Basel AG
Originally published by Birldlauser Verlag Basel1990
ISBN 978-3-7643-2530-5
OT47
Operator Theory: Advances and Applications
VoI.47
Editor:
I. Gohberg
Tel Aviv University
Ramat Aviv, Israel
Editorial Office:
School of Mathematical Sciences
Tel Aviv University
Ramat Aviv, Israel
Editorial Board:
A.'Atzmon (Tel Aviv) M. A. Kaashoek (Amsterdam)
J. A. Ball (Blacksburg) T. Kailath (Stanford)
L. de Branges (West Lafayette) H. G. Kaper (Argonne)
K. Clancey (Athens, USA) S. T. Kuroda (Tokyo)
L. A. Coburn (Buffalo) P. Lancaster (Calgary)
R. G. Douglas (Stony Brook) L. E. Lerer (Haifa)
H. Dym (Rehovot) E. Meister (Darmstadt)
A. Dynin (Columbus) B. Mityagin (Columbus)
P. A. Fillmore (Halifax) J. D. Pincus (Stony Brook)
C. Foias (Bloomington) M. Rosenblum (Charlottesville)
P. A. Fuhrmann (Beer Sheva) J. Rovnyak (Charlottesville)
S. Goldberg (College Park) D. E. Sarason (Berkeley)
B. Gramsch (Mainz) H. Widom (Santa Cruz)
J. A. Helton (La Jolla) D. Xia (Nashville)
D. Herrero (Tempe)
Honorary and Advisory
Editorial Board:
P. R. Halmos (Santa Clara) S. G. Mikhlin (Leningrad)
T. Kato (Berkeley) R. Phillips (Stanford)
P. D. Lax (New York) B. Sz.-Nagy (Szeged)
M. S. Livsic (Beer Sheva)
Birkhauser Verlag
Basel· Boston· Berlin
v
Table of Contents
Editorial Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
D. Alpay, 1. A. Ball, 1. Gohberg, L. Rodman
Realization and factorization for rational matrix functions with symmetries . . . . . 1
O. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1. Preliminaries .......................................... 4
2. Axiomatic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1 Admissible automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 General theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3. The associated group ..................................... 16
3.1 Realization theorems, the associated matrix, and examples. . . . . . . . . . 16
3.2 Factorization within the associated group . . . . . . . . . . . . . . . . . . . . . 19
4. Type I a symmetries ...................................... 23
5. Type I b symmetries ...................................... 27
6. Type II a symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
7. Type IIb symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
8. Degree preserving automorphisms of GL" (!R) ..................... 44
9. Continuous and analytic automorphisms of GL (JR) ... . . . . . . . . . . . . . . 55
m
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
D.Alpay, P. Dewilde, H. Dym
Lossless inverse scattering and reproducing kernels for upper triangular
operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2. Preliminaries .......................................... 69
3. A pair of transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4. An elementary Blaschke factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5. Lis and linear fractional transformations ... . . . . . . . . . . . . . . . . . . . . . . 90
6. An approximation problem ................................. 96
7. Reproducing kernel subspaces of Hilbert Schmidt operators . . . . . . . . . . . . . 102
8. Operator ranges ........................................ 113
9. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
1. A. Ball, M. Rakowski
Zero-pole stmcture of nonregular rational matrix functions . . . . . . . . . . . . . . . 137
o.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
1. Pole pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
2. Orthogonality inlRn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
3. Zero structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
4. Functions with a given local null-pole structure . . . . . . . . . . . . . . . . . . . . . 175
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
VI
H. Bercovici, C. Foias, A. Tannenbaum
Structured interpolation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
2. Classical commutant lifting theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
3. Remarks on the structured singular value ........................ 198
4. Block diagonal!:::, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
5. Structured singular values and dilations. . . . . . . . . . . . . . . . . . . . . . . . . . 208
6. Structured commutant lifting theorem .......................... 211
7. Structured Nevanlinna-Pick theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
8. Example ............................................. 216
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
D. A. Dritschel, 1. Rovnyak
Extension theorems for contraction operators on Krein spaces . . . . . . . . . . . . . 221
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
Chapter 1: Operator Theory on Krein Spaces ....................... 224
1.1 Definitions and preliminaries .......................... 224
1.2 Defect operators and Julia operators ..................... 234
1.3 Contraction and bicontraction operators ................... 238
1.4 Additional results on contractions and bicontractions ... . . . . . . . . 249
Chapter 2: Matrix Extensions of Contraction Operators ................ 254
2.1 The adjoint of a contraction ........................... 254
2.2 Column extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
2.3 Rowextensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
2.4 Two-by-two matrix completions . . . . . . . . . . . . . . . . . . . . . . . . . 267
Chapter 3: Commutant Lifting of Contraction Operators . . . . . . . . . . . . . . . . 273
3.1 Dilation theory ................................... 273
3.2 Commutant lifting ................................. 280
3.3 Characterization of extensions ......................... 285
3.4 Abstract Leech theorem ............................. 289
Appendix A: Complementation theory ........................... 292
Appendix B: More on Julia operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 301
VII
EDITORIAL INTRODUCTION
This volume consists of five papers which develop further the theory of extension and
interpolation of linear operators and matrix functions.
The first paper, "Realization and factorization for rational matrix functions with
symmetries," of D. Alpay, J.E. Ball, I. Gohberg, and 1. Rodman, is related to the problem
of homogeneous interpolation for rational matrix functions with symmetries. It contains a
description of properties of realization of a rational matrix function with given symmetries
and the solution of the inverse problem which is the interpolation problem of constructing
a symmetric rational matrix function from its null pole data. Factorization is also one
of the themes of this paper. The paper concludes with an analysis of different types of
symmetries.
The second paper, "Lossless inverse scattering and reproducing kernels for upper tri
angular operators," of D. Alpay, P. Dewilde, and H. Dym, is concerned with lossless inverse
scattering for nonstationary systems. The theory of point evaluations and Blaschke prod
ucts is developed for the nonstationary case. These tools will be used in future publications
on nonstationary generalizations of Nevanlinna Pick interpolation problems, where both
the interpolation points and the values consigned to these points will be block diagonal
operators.
The third paper, "Zero-pole structure of nonregular rational matrix functions," of
J .A. Ball and M. Rakowski, is concerend with homogeneous interpola*ion problems for
nonregular rational matrix functions. Here is described the null pole structure of such
matrix functions. The inverse problem, which is an interpolation problem of constructing
a rational matrix function from its null pole data, is also solved.
In the fourth paper, "Structured interpolation theory," of H. Bercovici, C. Foias and
A. Tannenbaum, is solved a Nevanllnna Pick type interpolation problem for matrix valued
functions. The novelty here is that instead of putting up restrictions on the norm of the
matrix function which is its singular value, a more sophisticated structured singular value
from systems theory is used. The latter is a generalization of the norm and the spectral
radius. The paper contains a new version of the commutant lifting theorem.
The fifth paper, "Extension theorems for contraction operators on Krein spaces," of
D.A. Dritschel and J. Rovnyak, starts with an introduction to operator theory on Krein
spaces and continues with general theorems on extensions of contractive operators. The
main theme is the theory of commutant lifting for contraction operators in Krein spaces.
One of the most eminent mathematicans of our time, Mark Grigorievich Krein passed
away in Odessa on October 17, 1989. Almost all of the authors submitted their papers with
a dedication to the memory of M.G. Krein, who made extremely important contributions
in the topics of this volume. The editor feels it very appropriate to follow this sponta
neous action and is joining the authors in dedicating the whole volume to the memory of
M.G. Krein as an expression of respect and admiration.
Operator Theory 1
Advances and Applications, Vol. 47
© 1990 Birkhiiuser Verlag Basel
REALIZATION AND FACTORIZATION FOR RATIONAL
MATRIX FUNCTIONS WITH SYMMETRIES
Daniel Alpay, Joseph A. BallI, Israel Gohberg and Leiba Rodman!
IN MEMORY OF MARK GRIGORIEVICH KREIN
In this paper we study rational matrix valued functions which are symmetric in various
senses. The problems of realization and minimal factorization are the main themes discussed for
these classes of functions.
O. INTRODUCTION.
Let F be a field which will be considered fixed thoughout the paper. By 'R we denote
the quotient field F(z) of the polynomial ring F[z] of F in the indeterminate Zj more concretely
EJ=-oo
one can view 'R as consisting of formal Laurent series fJ zi which have a presentation as
p(z) . q(z)-l with p and q polynomials. When the field is infinite, one can identify elements of'R
as functions from F into Fj for a finite field this point of view has the draw back that a nonzero
element of'R may correspond to the zero function. Let GLm('R) be the multiplicative group
of all regular (i.e. with determinant not equal to the zero element of'R) m X m matrices over
'Rj elements of GLm('R) thus more concretely may be thought of as formal Laurent series with
coefficients equal to matrices over F or, if F is infinite, as matrix (over F)-valued functions on F.
Given an automorphism A of this group GLm('R), we say that the rational matrix WE GLm('R)
= =
is A-symmetric if A(W) W. For example, let F a; and let A be the automorphism on
GLm('R) defined by
A(W)(z) = W(-z)*-l.
Then the A-symmetric rational matrices are exactly those rational matrix functions with unitary
values on the imaginary line. In general, our main aim is to describe all A-symmetric rational
matrices and secondly, to describe all factorizations of A-symmetric rational matrices with fac
tors which are also A-symmetric. We are also interested in the problem of characterizing the
elementary A-symmetric matrices, i.e. those A-symmetric matrices with no nontrivial represen
tation as a product of A-symmetric matrices. These three problems can also be considered for
rational matrices with respect to several symmetries.
=
In this paper we will be especially interested in the case when F a; and the automor-
1 Research of these authors was partially supported by grants from the National Science Foundation.
2 Alpay et al.
phism A has one of the following four forms:
=
(Ia) A(W)(z) M-1W 0 cp(z)M
(Ib) A(W)(z) = M-1W 0 cp(z)M
=
(ITa) A(W)(z) M-1[W 0 cp(z)]-IT M
(IIb) A(W)(z) = M-1[W 0 cp(z)]-h M
where M is a fixed invertible matrix and cp is a Mobius transformation. Our main approach to
solving this problem is the representation of proper rational matrices in the form
(1.1) W(z) = D + C(zI - A)-IB
where A, B, C and D are finite matrices of appropriate sizes. Such a representation for W( z) we
refer to as a realization of W(z).
All four types of automorphisms mentioned above belong to the class of automorphisms
which have the following property: there exists a map (A, B, C, D) ..... (AA, BA, CA, DA) such
that
if
W(z) = D + C(zI - A)-l B
whenever both W(z) and A(W)(z) are analytic and invertible at infinity. Here we work only
with minimal realizations, i.e., we assume that the (square) size of A is as small as possible
among all presentations ofW(z) in the form (1.1). We also assume that the map (A,B,C,D) .....
(AA, BA, CA, DA) satisfies some additional properties which hold for the types of automorphisms
listed above.
Namely, we demand that
L'[ L,[
([ ~l BA~2 BB~2 C1 D1C2]A' (DID2)A) =
([ AOA B~~2A], [ B~~2A ] , [CIA D1AC2A], (D1AD2A)) .
and, for T E a:;nxn there exists a map T ..... TA (called the associated map to A) such that
In each of the four types of examples listed above these four properties are easily verified. For
any automorphism A which satisfies these properties, we classify the A-symmetric matrices
