Table Of ContentOperatorTheory: Advances and
Applications
Vol. 134
Editor:
I. Gohberg
Editorial Office:
School of Mathematical
Sciences
Tel Aviv University
Ramat Aviv, Israel
Editorial Board: P. Lancaster(Calgary)
J. Arazy (Haifa) L.E. Lerer (Haifa)
A. Atzmon (Tel Aviv) B. Mityagin (Columbus)
J. A. Ball (Blacksburg) V. V. Peller(Manhattan, Kansas)
A. Ben-Artzi (Tel Aviv) J. D. Pincus (Stony Brook)
H. Bercovici (Bloomington) M. Rosenblum (Charlottesville)
A. Bottcher(Chemnitz) J. Rovnyak (Charlottesville)
K. Clancey (Athens, USA) D. E. Sarason (Berkeley)
L. A. Coburn (Buffalo) H. Upmeier(Marburg)
K. R. Davidson (Waterloo, Ontario) S. M. Verduyn Lunel (Amsterdam)
R. G. Douglas (Stony Brook) D. Voiculescu (Berkeley)
H. Dym (Rehovot) H. Widom (Santa Cruz)
A. Dynin (Columbus) D. Xia (Nashville)
P. A. Fillmore (Halifax) D. Yafaev (Rennes)
P. A. Fuhrmann (BeerSheva)
S. Goldberg{College Park) Honorary and Advisory
B. Gramsch (Mainz) Editorial Board:
G. Heinig (Chemnitz) C. Foias (Bloomington)
J. A. Helton (La Jolla) P. R. Halmos (Santa Clara)
M.A. Kaashoek (Amsterdam) T. Kailath (Stanford)
H.G. Kaper(Argonne) P. D. Lax (New York)
S.T. Kuroda (Tokyo) M. S. Livsic (Beer Sheva)
Interpolation Theory,
Systems Theory
and Related Topics
The Harry Dym Anniversary Volume
Daniel Alpay
Israel Gohberg
Victor Vinnikov
Editors
Springer Basel AG
Editors:
Daniel Alpay Victor Vinnikov
Department of Mathematics Department of Mathematics
Ben-Gurion University of the Negev Ben-Gurion University of the Negev
P.O. Box 653 P.O. Box 653
Beer Sheva 84105 Beer Sheva 84105
Israel Israel
e-mail: [email protected] e-mail: [email protected]
Israel Gohberg
School of Mathematical Sciences
Raymond and Beverly Sackler
Faculty of Exact Sciences
Tel Aviv University
Ramat Aviv 69978
Israel
e-mail: [email protected]
2000 Mathematics Subject Classification 47-06; 47A57, 47N20, 47N70, 65D05, 65E05
A CIP catalogue record for this book is available from the
Library of Congress, Washington D.C., USA
Deutsche Bibliothek Cataloging-in-Publication Data
lnterpolation theory, systems theory and related topies : the Harry Dym
anniversary volume / Daniel Alpay ... ed. - Basel ; Boston; Berlin:
Birkhăuser, 2002
(Operator theory ; Val. 134)
ISBN 978-3-0348-9477-7 ISBN 978-3-0348-8215-6 (eBook)
DOI 10.1007/978-3-0348-8215-6
ISBN 978-3-0348-9477-7
This work is subject to copyright. Al! rights are reserved, whether the whole ar part of the material is concerned,
specifically the rights of translation, reprinting, re-use of il1ustrations, recitation, broadcasting, reproduction on microfilms
ar in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.
© 2002 Springer Basel AG
Original1y published by Birkhăuser Verlag in 2002
Softcover reprint of the hardcover 1s t edition 2002
Printed on acid-free paper produced from chlorine-free pulp. TCF =
Cover design: Heinz Hiltbrunner, Basel
ISBN 978-3-0348-9477-7
987654321 www.birkhauser-science.com
Contents
Editorial Introduction Vll
Acknowledgments ............................. xi
Portrait of Harry Dym .................................... xii
H. Dym
Looking Back ................................... 1
List ofPublications ofHarry Dym 19
1. Gohberg
On Joint Work with Harry Dym 25
J. Rovnyak
Methods ofKreIn Space Operator Theory................ 31
D. Alpay, T. Constantinescu, A. Dijksma, and J. Rovnyak
Notes on Interpolation in the Generalized Schur Class.
I. Applications ofRealization Theory 67
D.Z. Arov (with an appendix by D.Z. Arov and J. Rovnyak)
Stable Dissipative Linear Stationary Dynamical
Scattering Systems 99
J.A. Ball, K.F. Clancey, and V. Vinnikov
Concrete Interpolation ofMeromorphic Matrix Functions
on Riemann Surfaces 137
M.F. Bessmertnyz
On Realizations ofRational Matrix Functions
of Several Complex Variables 157
C.S. Calude and B. Pavlov
The Poincare-HardyInequality on the Complement
ofa Cantor Set 187
1. Gohberg, M.A. Kaashoek, and F. van Schagen
Finite Section Method for Linear Ordinary Differential Equations
on the Full Line 209
D. Hershkowitz
On the Spectral Radius ofMulti-Matrix Functions.... 225
V. Katsnelson
A Generic Schur Function is an Inner One 243
vi Contents
A. Kheifets
Abstract Interpolation Scheme for Harmonic Functions 287
M.S. Livsic
Chains ofSpace-Time Open Systems and DNA...................... 319
A.C.M. Ran and L. Rodman
A Class ofRobustness Problems in Matrix Analysis 337
L. Sakhnovich
Dual Discrete Canonical Systems and
Dual Orthogonal Polynomials .................................. 385
V. Tkachenko
Non-Selfadjoint Sturm-Liouville Operators with
Multiple Spectra..................................................... 403
Editorial Introduction
This volume is based on the proceedings ofthe Toeplitz Lectures 1999 and ofthe
Workshop in Operator Theory held in March 1999 at Tel-Aviv University and
at the Weizmann Institute of Science. The workshop was held on the occasion
of the 60th birthday of Harry Dym, and the Toeplitz lecturers were Harry Dym
and Jim Rovnyak. The papers in the volume reflect Harry's influence on the field
ofoperator theory and its applications through his insights, his writings, and his
personality. The volume begins with an autobiographical sketch, followed by the
list ofpublications ofHarryDym and the paper ofIsrael Gohberg: On Joint Work
with Harry Dym.
The following paper by Jim Rovnyak: Methods ofKrdn Space Operator The
ory, is based on his Toeplitz lectures. It gives a survey ofold and recents methods
of KreIn space operator theory along with examples from function theory, espe
cially substitution operators on indefinite Dirichlet spaces and their relation to
coefficient problems for univalent functions, an idea pioneered by 1. de Branges
and underlying his proofofthe Bieberbach conjecture (see [9]).
The remaining papers (arranged in the alphabetical order) can be divided
into the following categories.
Schur analysis and interpolation
In Notes on Interpolation in the Generalized Schur Class. I, D. Alpay, T. Con
stantinescu, A. Dijksma, and J. Rovnyak use realization theory for operator colli
gations in Pontryagin spaces to study interpolation and factorization problems in
generalized Schur classes.
In his paper A Generic Schur Function Is an Inner One, V. Katsnelson uses
theSchurparameterstoputa probabilitymeasureonthesetofallSchurfunctions,
and studies the genericity of inner functions by the methods of multiplicative
ergodic theory.
A. Kheifets, AbstractInterpolation Scheme for Non Analytic Problems, devel
opsa generalizationoftheabstract interpolation problemofKatsnelson-Kheifetz
Yuditskii (see [14, 15]) to handle non analytic interpolation problems such as
the Nehari interpolation problem. One of the key ideas is a systematic replace
ment of unitary colligations, or equivalently conservative input/state/output sys
tems, by generally non-orthogonal (non-causal) scattering systems as introduced
by Adamyan-Arov [1].
Several complex variables and Riemann surfaces
In Concrete Interpolation ofMeromorphic Matrix Functions on RiemannSurfaces,
J.A. Ball, K.F. Clancey, and V. Vinnikov investigate the problems ofinterpolat
ing matrix pole-zero data with multiple-valued meromorphic matrix functions on
viii Editorial Introduction
compact Riemann surfaces. This is related on the one hand to homogeneous in
terpolation problems for rational matrix functions as studied in [5], and on the
other hand to the study ofvector bundles on compact Riemann surfaces initiated
by Andre Weil in [18] and actively pursued in the last two decades in algebraic
geometry (see, e.g., [17]).
The paper by M.F. Bessmertnyr, On Realizations of Rational Matrix Func
tions ofSeveral Complex Variables, is an English translation, prepared by Daniel
Alpay and Victor Katsnelson, of a part of a Ph. D. thesis that was written in
Russian in 1982 and has never been published. It deals with realization theory for
rational matrix functions ofseveral complexvariables, especially for functions sat
isfying positivity conditions; its publication now is especially suitable because of
a recent surge ofactivity in the area- the works ofAgler-McCarthy [4], Alpay
Kaptanoglu [2], Ball-Sadosky-Vinnikov [6], Ball-Trent [7], Kalyuzhniy [12, 13]
inspired by the work ofAgler [3].
Matrix theory
The paper by D. Hershkowitz, On the Spectral Radius ofMulti-Matrix Functions,
deals with the behaviour of the spectral radius of a matrix with positive entries
under multivariable matrix functions, and some other related questions. Mostly a
survey, it contains also original results.
A Class ofRobustness Problems in Matrix Analysis, by A. Ran and L. Rod
man, is a survey of a class of perturbation problems that has been extensively
studied by the authors and their collaborators over a period ofseveral years. The
stage is set by posing an abstract "metaproblem" followed by a careful review of
results concerning the pervasive question ofthe stability ofinvariant subspaces.
System theory
The main part ofthe paper Stable Dissipative Linear Stationary Dynamical Scat
tering Systems byD.Z. Arov is an English translation, prepared by D.Z. Arovand
J. Rovnyak, ofa highly influential article originally published in Russian in 1979;
it deals with (linear time-invariant) dissipative input/state/output systems, and
their role in electrical networks (Darlington synthesis), operator theory, and func
tion theory. There are two new appendices, the first one by D.Z. Arov providing a
commentary and an update ofthe results, and the second one by D.Z. Arov and
J. Rovnyak showing some directions for generalizations and further development.
In Chains of Space-Time Open Systems and DNA, M.S. Livsic discusses a
striking resemblance between chains of overdetermined multidimensional (space
time) systems, that appear in the spectral analysis oftuples ofnonselfadjoint and
nonunitaryoperators [16], and chainsofnucleotidesinmolecular biology. Heshows
that some important properties of the DNA can be given a natural explanation
using the methods ofsystem theory.
Editorial Introduction ix
Differential equations and mathematical physics
The paper Dual Discrete Canonical SystemsbyL. Sakhnovichdiscusses thenotion
of dual canonical systems in the discrete case. The notion was introduced in the
continuous case in a recent paper of Dym and Sakhnovich [10], generalizing (in
that case) the notion of dual string equations which was introduced by Kac and
KreIn for scalar strings [11].
In Finite Section Method for Linear Ordinary Differential Equations on the
Full Line,I. Gohberg, M.A. Kaashoek, and F.vanSchagenstudysolutionsoflinear
ordinarydifferentialequationsonthefull lineaslimitsofsolutionsofcorresponding
equations on smaller intervals (with appropriate boundary or initial conditions).
Both the time-invariant and the time-varying cases are considered.
C. Calude and B. Pavlov, The Poincare-Hardy Inequality on the Comple
ment ofa Cantor Set, derive the Poincare-Hardy inequality (an important tool in
classical analysis, as well as in quantum mechanics, mathematical hydrodynamics,
and quantum scattering) in ~3 on the complement ofa Cantor set. The approach
to the problemis viaa certainrelevant dynamicalsystem, inspired by Carleson [8].
Non-Selfadjoint Sturm-Liouville Operators with Multiple Spectra by V. Tka
chenko is related to the spectral theory of non-selfadjoint Sturm-Liouville opera
tors. Whileit wasgenerallybelievedthat anoperatorwitha complexpotentialcan
have spectral points ofan arbitrary multiplicity, not a single explicit example of,
say, operator on a finite interval with multiple Dirichlet or Neumann spectra was
previously known. Among other results, this paper constructs a Sturm-Liouville
operator with an arbitrary given (symmetric) Dirichlet spectrum An subject only
to a restriction dealing with a suitable asymptotic behavior of An.
References
[1] V.M. Adamyan and D.Z. Arov, On unitary coupling of semiunitary operators,
Matern. Issled. 1 (1966), 3-64 (Russian); translated in Amer. Math. Soc. Transl.
Ser. 295 (1970), 75-129.
[2] D. Alpay and H.T. Kaptanoglu, Sous espaces de codimension finie dans la boule
uniteetunprobleme de factorisation, C. R. Acad. Sci. ParisSer. I Math. 331 (2000),
947-952.
[3] J. Agler, On the representation ofcertain holomorphic functions defined on a poly
disc, Topics in Operator Theory: Ernst D. Hellinger Memorial Volume (L. de
Branges, I. Gohberg, and J. Rovnyak, eds.), Operator Theory: Adv. Appl., vol. 48,
Birkhauser Verlag, Basel, 1990, pp. 47-66.
[4] J. Agler and J.E. McCarthy, Nevanlinna-Pick interpolation on the bidisk, J. Reine
Angew. Math. 506 (1999), 191-204.
[5] J.A. Ball, I. Gohberg, and L. Rodman, Interpolation ofrational matrix functions,
Operator Theory: Adv. Appl., vol. 45, Birkhauser Verlag, Basel, 1990.
[6] J.A. Ball, C. Sadosky, and V. Vinnikov, Scattering systems with several evolutions
and multidimensional input/state/output systems, preprint.
x Editorial Introduction
[7] J.A. BallandT.T. Trent, Unitary colligations, reproducingkernel Hilbert spaces, and
Nevanlinna-Pick interpolation in several variables, J. Funct. Anal. 157 (1998), 1-61.
[8] L. Carleson, Selected problems on exceptional sets, Van Nostrand, Princeton, 1967.
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1986), Amer. Math. Soc., Providence, 1988, pp. 25-42.
[10] H. Dym and L.A. Sakhnovich, On dual canonical systems and dual matrix string
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[11] I.S. Kac and M.G. Krein, On the spectralfunctions ofthe string, Russiantranslation
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Supplement II, pp. 648-737 (Russian); translated in Amer. Math. Soc. Transl. (2)
103 (1974), 19-102.
[12] D.S. Kalyuzhniy, Multiparametric dissipative linear stationary dynamical scattering
systems: discrete case, J. Operator Theory 43 (2000), no. 2, 427-460.
[13] , Multiparametric dissipative linear stationary dynamical scattering systems:
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[14] V.E. Katsnelson, A.Y. Kheifets, and P.M. Yuditskii, An abstract interpolation prob
lem andthe theory ofextensionsofisometricoperators,OperatorsinFunctionSpaces
and Problems in Function Theory (V. A. Marchenko, ed.), Naukova Dumka, Kiev,
1987, pp. 83-96, 146 (Russian); translated in Topics in Interpolation Theory (Harry
Dymetal., editors), Operator Theory: Adv. Appl. 95, BirkhiiuserVerlag, Basel, 1997,
pp. 283-298.
[15] A. Kheifets, The abstract interpolation problem and applications, Holomorphic
SpacesandTheirOperators(S. Axler, J. McCarthy,andD.Sarason,eds.),Math. Sci.
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[16] M.S. Livsic, N. Kravitsky, A.S. Markus, andV. Vinnikov, Theory ofcommutingnon
selfadjointoperators,MathematicsandItsApplications,vol. 332,Kluwer, Dordrecht,
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Daniel Alpay, Israel Gohberg, Victor Vinnikov
