Table Of ContentOperator Theory
Advances and Applications
Vol. 96
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I. Gohberg
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A. Ben-Artzi (Tel Aviv) J. Rovnyak (Charlottesville)
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Schur Functions, Operator
Colligations, and Reproducing
Kernel Pontryagin Spaces
Daniel Alpay
Aad Dijksma
James Rovnyak
Hendrik de Snoo
Springer Basel AG
Authors:
Daniel Alpay James Rovnyak
Department of Mathematics Department of Mathematics
Ben-Gurion University of the Negev University of Virginia
P.O.Box 653 Charlottesville, VA 22903-3199
84105 Beer-Sheva USA
Israel e-mail: [email protected]
e-mail: [email protected]
AadDijksma Hendrik de Snoo
Department of Mathematics Department of Mathematics
University of Groningen University of Groningen
P.O.Box 800 P.O.Box 800
9700 A V Groningen 9700 A V Groningen
The Ne therlands The Netherlands
e-mail: [email protected] e-mail: [email protected]
1991 Mathematics Subject Classification 47A48, 47B50 (primary), 46C20, 46E22, 47A45 (secondary)
A CIP catalogue record for this book is available from the
Library ofCongress, Washington D.C., USA
Deutsche Bibliothek Cataloging-in-Publication Data
Schur functions, operator colligations and reproducing kernel
Pontryagin spaces I Daniel Alpay ... - Basel ; Boston; Berlin:
B irkhiiuser, 1997
(Operator theory ; VoI. 96)
ISBN 978-3-0348-9823-2 ISBN 978-3-0348-8908-7 (eBook)
DOI 10.1007/978-3-0348-8908-7
This work is subject to copyright. AII rights are reserved, whether the whole or part of the material is concerned,
specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms
or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.
© 1997 Springer Basel AG
Originally published by Birkhiiuser Verlag in 1997
Softcover reprint ofthe hardcover 1s t edition 1997
Printed on acid-free paper produced from chlorine-free pulp. TCF
00
Cover design: Heinz Hiltbrunner, Basel
ISBN 978-3-0348-9823-2
987654321
Dedicated to Heinz Langer
on the occasion of his 60-th birthday,
in appreciation, admiration, and amity.
CONTENTS
Introduction ........................................................... IX
Chapter 1: Pontryagin Spaces and Operator Colligations
1.1 Reproducing kernel Pontryagin spaces ............................. 1
1.2 Operator colligations .............................................. 13
1.3 Julia operators and contractions ................................... 19
1.4 Extension of densely defined linear relations ....................... 27
1.5 Complementation and reproducing kernels
A. Complementation in the sense of de Branges ................... . 31
B. Applications to reproducing kernel Pontryagin spaces 36
Chapter 2: Schur Functions and their Canonical Realizations
2.1 Pontryagin spaces Sj(S), Sj(S), and :!J(S) .......................... 41
2.2 Canonical coisometric and isometric realizations ................... 49
2.3 Canonical unitary realization ...................................... 54
2.4 Unitary dilations of coisometric and isometric colligations. . . . . . . . . . . 62
2.5 Classes S",(J, cB)
A. Definition and basic properties ................................. 68
B. Conform ally invariant view ..................................... 71
C. Application to factorization of operator-valued functions ........ 79
D. A non-holomorphic kernel .... '" ... ... ... ... ... .. . .. .. . .. . . . .. . 82
Chapter 3: The State Spaces
3.1 Invariance under difference quotients .............................. 83
3.2 Spaces Sj(S) ....................................................... 88
3.3 Spaces Sj (S) ....................................................... 99
3.4 Spaces :!J(S) ...................................................... 106
viii CONTENTS
3.5 Examples and miscellaneous results
A. Rational unitary functions ...................................... 113
B. Symmetry in the state spaces ................................... 119
C. Some consequences of Leech's theorem.......................... 122
D. The scalar case: 8(z) E $)(8) if and only if 8(z) E $)(8) ......... 127
Chapter 4: Structural Properties
4.1 Factorization and invariant subspaces
A. Inclusion of spaces $)(8) ........................................ 129
B. Inclusion of spaces 1'(8) ........................................ 138
4.2 KreIn-Langer factorization
A. Existence and properties....................................... 141
B. Strongly regular representations ................................ 150
4.3 The Potapov-Ginzburg transform .................................. 155
4.4 Applications to the realization theory
A. KreIn space inner and outer spaces ~ and ~ .................... 163
B. Other base points .............................................. 174
C. Examples ...................................................... 177
4.5 Canonical models ................................................. 179
Epilogue: Open Questions and Directions for Further Work............. 185
Appendix: Some Finite-Dimensional Spaces............................. 191
Notes.................................................................. 203
References ............................................................. 211
Notation Index......................................................... 221
Author Index .......................................................... 223
Subject Index .......................................................... 225
INTRODUCTION
The Schur class in complex analysis is the set of holomorphic functions
which are defined and bounded by one on the unit disk. It occurs prominently
in interpolation theory and invariant subspace theory as well as in applications
areas. Certain kernels induced by a given Schur function recur frequently. They
are reproducing kernels for functional Hilbert spaces, which today we understand
to be state spaces for canonical coisometric, isometric, and unitary colligations
whose characteristic functions coincide with the given Schur function.
Operator generalizations of the Schur class consist of functions 8 (z) defined
on a subregion D(8) of the unit disk containing the origin whose values are
operators in £(J, Qj) for some Hilbert, Pontryagin, or KreIn spaces J and Qj. We
associate with such functions 8 (z) the three kernels
_ 1-8(z)8(w)*
K ( )
s w, z - 1 - ,
-zw
K-( ) _ 1 - 8(z)8(w)*
s W,z - 1 - ,
-zw
8(z) - ~(w) )
Ks(w,z)
z-w
( ,
Ds(w, z) = 8(z) _ ~(w)
Ks(w, z)
z-w
where 8(z) = 8(z)* and 1 denotes either the scalar unit or an identity opera
tor, depending on context. When these kernels are nonnegative, they are repro
ducing kernels for Hilbert spaces 5)(8), 5)(8), 1)(8) of vector-valued functions.
The spaces appear in a canonical model for contraction operators given by L.
de Branges and J. Rovnyak in the case that J and Qj are Hilbert spaces. More
generally, we use the hypothesis that the three kernels have negative squares
/'l,
for some nonnegative integer In this case, we say that 8 (z) belongs to the gen
/'l,.
eralized Schur class S",(J, Qj). According to a theory founded by L. Schwartz and
P. Sorjonen, spaces 5)(8), 5)(8), 1)(8) still exist but now as Pontryagin spaces
having the negative index It should be noted that indefiniteness enters the
/'l,.
subject also in another way, namely, when J and Qj are not Hilbert spaces but
Pontryagin or KreIn spaces, a situation pioneered by V. P. Potapov in the matrix
case.
The indefinite cases have been studied in a number of places, notably in
a series of papers by M. G. KreIn and H. Langer and in more recent works by
L. de Branges. The KreIn-Langer theory assumes that the spaces J and Qj are
Hilbert spaces and is motivated by spectral theory, classical representations of
x INTRODUCTION
resolvents, and function-theoretic questions. The theory of de Branges adopts
a systems viewpoint and uses a notion.of complementation to make the key
constructions. Although many elegant results are obtained in these and other
sources in the literature, the indefinite theory overall is less complete than the
Hilbert space cases.
The purpose of this work is to present a theory of the generalized Schur
classes SK(~' Q;) in typical indefinite situations. Our main tool is a representation
of the spaces jj(8), jj(S), 1)(8) as the state spaces of canonical coisometric, iso
metric, and unitary colligations. The spaces ~ and Q; are allowed to be indefinite,
but in most of the results it is assumed that they are Pontryagin spaces having
the same negative index. However, we make no assumptions on the positive in
dices of ~ and Q;, and so our notion of generalized Schur function permits 8(z)
to have a "rectangular" form. The main results can be interpreted for matrix
or scalar-valued functions. Our methods combine the theory of linear relations,
reproducing kernel Pontryagin spaces, and abstract operator theory. This ap
proach contains elements of the KreIn-Langer and de Branges viewpoints but is
different from both.
When all of the spaces ~ and Q; and jj(8), jj(S), 1)(8) are Hilbert spaces,
our account is an exposition of known results in a setting of colligations and
realization theory. The Hilbert space case in turn guides the indefinite theory. In
brief outline:
• Chapter 1 is devoted to preliminary notions concerning reproducing kernel
Pontryagin spaces, linear relations, and complementation.
• In Chapter 2 we make the main construction of canonical colligations and
introduce the generalized Schur classes SK(J, Q;).
• Chapter 3 details the properties of the state spaces and gives examples,
which include rational unitary functions.
• In Chapter 4 we explore the relationship between invariant subspaces and
factorization and obtain information for the indefinite case from the Hilbert
space case by means of the Potapov-Ginzburg transform. The main trans
formations in state spaces are used to construct canonical models.
• In an epilogue we note some questions that are left unresolved and possible
areas for future development.
• An appendix derives properties of finite-dimensional spaces which illustrate
the theory and are used in our derivation of the KreIn-Langer factorization
in Chapter 4.
As to our motivation to produce this work, chiefly it seems a natural exten
sion of existing theories. At the same time, we note that there is much current
interest in indefinite inner products in interpolation problems, systems theory,
model theory, and even univalent functions. It is well known that such problems
INTRODUCTION xi
present essential new difficulties beyond the Hilbert space case, and many issues
are unresolved at this time. The main conclusion of this work is that the colliga
tion and state space structures associated with generalized Schur functions are
well behaved in the indefinite case. It is hoped that this might prove useful in
the pursuit of the open problems in the indefinite case.
The approach developed here was announced in Alpay, Dijksma, Rovnyak,
and de Snoo [1996]. An expository account for scalar-valued functions is given
in Alpay, Dijksma, Rovnyak, and de Snoo [1997].
Acknowledgements
The Ben-Gurion University of the Negev, University of Groningen, and
University of Virginia supported exchange visits of the co-authors over the years
during which this work was carried out. James Rovnyak was Dozor Visiting
Professor at the Ben-Gurion University of the Negev in the summer of 1996, and
he was supported by the National Science Foundation under the grant DMS-
9501304 and at the Mathematical Sciences Research Institute (Berkeley) under
the grant DMS-9022140.
