Operator Theory, System Theory and Related Topics: The Moshe Livšic Anniversary Volume

567 Pages·2001·13.86 MB·English

by  Daniel Alpay

Operator Theory: Advances and
Applications
Vol.123
Editor:
I. Gohberg
Editorial Office:
School of Mathematical H.G. Kaper (Argonne)
Sciences ST. Kuroda (Tokyo)
Tel Aviv University P. Lancaster (Calgary)
Ramat Aviv, Israel L.E. Lerer (Haifa)
E. Meister (Darmstadt)
Editorial Board: B. Mityagin (Columbus)
J. Arazy (Haifa) V. V. Peller (Manhattan, Kansas)
A. Atzmon (Tel Aviv) J. D. Pincus (Stony Brook)
J. A. Ball (Blacksburg) M. Rosenblum (Charlottesville)
A. Ben-Artzi (Tel Aviv) J. Rovnyak (Charlottesville)
H. Bercovici (Bloomington) D. E. Sarason (Berkeley)
A. Böttcher (Chemnitz) H. Upmeier (Marburg)
K. Clancey (Athens, USA) S. M. Verduyn-Lunel (Amsterdam)
L. A. Coburn (Buffalo) D. Voiculescu (Berkeley)
K. R. Davidson (Waterloo, Ontario) H. Widom (Santa Cruz)
R. G. Douglas (Stony Brook) D. Xia (Nashville)
H. Dym (Rehovot) D. Yafaev (Rennes)
A. Dynin (Columbus)
P. A. Fillmore (Halifax) Honorary and Advisory
P. A. Fuhrmann (Beer Sheva) Editorial Board:
S. Goldberg (College Park) C. Foias (Bloomington)
B. Gramsch (Mainz) P. R. Haimos (Santa Clara)
G. Heinig (Chemnitz) T. Kailath (Stanford)
J. A. Helton (La Jolla) P. D. Lax (New York)
M.A. Kaashoek (Amsterdam) M. S. Livsic (Beer Sheva)
Operator Theory,
System Theory and
Related Topics
The Moshe Livsic Anniversary Volume
D. Alpay
V. Vinnikov
Editors
Springer Basel AG
Editors:
Daniel Alpay
Victor Vinnikov
Department of Mathematics
Ben-Gurion University of the Negev
P.O. Box 653
Beer Sheva 84105
Israel
2000 Mathematics Subject Classification 47-06; 30-06, 93-06
A CIP catalogue record for this book is available from the
Library of Congress, Washington D.C., USA
Deutsche Bibliothek Cataloging-in-Publication Data
Operator theory, system theory and related topics : the Moshe Livsic anniversary volume /
D. Alpay ; V. Vinnikov ed.. - Basel; Boston ; Berlin : Birkhäuser, 2001
(Operator theory ; Vol. 123)
ISBN 978-3-0348-9491-3 ISBN 978-3-0348-8247-7 (eBook)
DOI 10.1007/978-3-0348-8247-7
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© 2001 Springer Basel AG
Originally published by Birkhäuser Verlag, Basel in 2001
Member of the BertelsmannSpringer Publishing Group
Printed on acid-free paper produced from chlorine-free pulp. TCF °°
Cover design: Heinz Hiltbrunner, Basel
Contents
Editorial Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Vll
Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
List of Publications of M. S. Livsic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
M. S. LIVSIC
Vortices of 2D Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
D. ALPAY, A. DIJKSMA, J. ROVNYAK, AND H. DE SNOO
Realization and Factorization in Reproducing Kernel Pontryagin Spaces. . .. 43
S. S. BOIKO, V. K. DUBOVOY, B. FRITSCHE, AND B. KIRSTEIN
Models of Contractions Constructed from the Defect Function of Their
Characteristic Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 67
S. S. BOIKO, V. K. DUBOVOY, AND A. JA. KHEIFETS
Measure Schur Complements and Spectral Functions of Unitary Operators
with Respect to Different Scales ......................................... 89
V. BOLOTNIKOV
On the Second Order Interpolation for Rational Vector Functions. . . . . . . . .. 139
M. BRISKIN, J.-P. FRAN<;:OISE, AND Y. YOMDIN
Generalized Moments, Center-focus Conditions, and Compositions of
Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 161
M. CHUGUNOVA
Inverse Spectral Problem for the Sturm-Liouville Operator with
Eigenvalue Parameter Dependent Boundary Condition .................... 187
M. DRITSCHEL
A Module Approach to Commutant Lifting on KreIn Spaces ............... 195
H. DYM AND L. A. SAKHNOVICH
On Dual Canonical Systems and Dual Matrix String Equations . . . . . . . . . . .. 207
S. FEDOROV
Angle between Subspaces of Analytic and Antianalytic Functions
in Weighted L2 Spaces on the Boundary of a Multiply
Connected Domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 229
B. FREYDIN
Bitangential Interpolation for Upper Triangular Operators with
Infinitely Many Interpolation Nodes ...................................... 257
VI Contents
F. GESZTESY, N. J. KALTON, K. A. MAKAROV, AND E. TSEKANOVSKII
Some Applications of Operator-valued Herglotz Functions ................. 271
Ju. P. GINZBURG
Analogues of a Theorem of Frostman on Linear Fractional
Transformations of Inner Functions and the Typical Spectral Structure
of Analytic Families of Weak Contractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 323
V. KATSNELSON
Right and Left Joint System Representation of a
Rational Matrix Function in General Position
(System Representation Theory for Dummies) ............................ 337
V. KHATSKEVICH, S. REICH, AND D. SHOIKHET
One-parameter Semigroups of Fractional-linear Transformations ........... 401
P. KURASOV
Scattering from an Impurity: Lax-Phillips Approach. . . . . . . . . . . . . . . . . . . . . .. 413
M. S. MACCORMICK AND B. S. PAVLOV
Spectral Theory of Wiener-Hopf Operators and Functional Model .......... 433
M. PUTINAR
Operator Dilations with Prescribed Commutators. . . . . . . . . . . . . . . . . . . . . . . .. 453
1. A. SAKHNOVICH
Condition of Orthogonality of Spectral Matrix Function. . . . . . . . . . . . . . . . . .. 461
A. V. STRAUSS
Functional Models of Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 469
V. TKACHENKO
Non-self-adjoint Periodic Dirac Operators ................................ 485
D. YAKUBOVICH
A Note on Hyponormal Operators Associated with
Quadrature Domains .................................................... 513
P. YUDITSKII
Two Remarks on Fuchsian Groups of Widom Type. . . . . . . . . . . . . . . . . . . . . . .. 527
V. A. ZOLOTAREV
A Functional Model for the Lie Algebra SL(2, lR) of Linear
Non-self-adjoint Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 539
Editorial Introduction
This volume presents the refereed proceedings of the Conference in Operator The
ory in Honour of Moshe Livsic 80th Birthday, held June 29 to July 4, 1997, at the
Ben-Gurion University of the Negev (Beer-Sheva, Israel) and at the Weizmann In
stitute of Science (Rehovot, Israel). The volume contains papers in operator theory
and its applications (understood in a very wide sense), many of them reflecting,
directly or indirectly, a profound impact of the work of Moshe Livsic.1
Moshe (Mikhail Samuilovich) Livsic was born on July 4, 1917, in the small town
of Pokotilova near Uman, in the province of Kiev in the Ukraine; his family moved
to Odessa when he was four years old. In 1933 he enrolled in the Department
of Physics and Mathematics at the Odessa State University, where he became a
student of M. G. Krein and an active participant in Krein's seminar - one of the
centres where the ideas and methods of functional analysis and operator theory
were being developed. Besides M. G. Krein, M. S. Livsic was strongly influenced
by B. Va. Levin, an outstanding specialist in the theory of analytic functions. A
deep understanding of operator theory as well as function theory and a penetrating
search of connections between the two, were to become one of the landmarks of
M. S. Livsic's work.
M. S. Livsic defended his Ph. D. thesis in 1942 at the University of Odessa
which was evacuated to Maikop in the beginning of the war. In the thesis he had
introduced a generalized problem of moments and solved it using the theory of
Hermitian operators in a Hilbert space. Both the problem itself and the method
of solution have started a long line of investigations, including some well-known
works of M. G. Krein. The generalized problem of moments and Livsic's theorem
are discussed in the article of M. Briskin, J.-P. Fran<;oise, and Y. Yomdin in this
volume.
In 1943-1944 M. S. Livsic discovered the notion of the characteristic function.
Following the completion by J. von Neumann of the spectral theory of selfadjoint
ITwo other volumes in the series Operator Theory: Advances and Applications dedicated to
Moshe Livsic are Topics in Operator Theory and Interpolation (Essays dedicated to M. S. Livsic
on the occasion of his 70th birthday), OT29 (1988), and Nonselfadjoint Operators and Related
Topics (Workshop on Operator Theory and Its Applications, Beersheva, February 24-28, 1992),
OT73 (1994). The first one of these contains a biography of Moshe Livsic by H. Dym, 1. Gohberg,
and N. Kravitsky that supplements a sketch of the scientific path of Moshe Livsic below. It also
contains a list of publications of Moshe Livsic till 1987; unfortunately this list does not include
references to English translations of papers published in Russian ~r to Mathematical Reviews,
so we have decided to include a complete list of publications of Moshe Livsic up to the present
(with all the relevant information) in this volume as well. For additional biographical details see
also the reminiscences of Moshe Livsic about V. P. Potapov in Topics in Interpolation Theory,
OT95 (1997).
viii Editorial Introduction
operators in the 1930's, the issue of constructing a spectral theory for nonselfad
joint operators was enthusiastically discussed. Alternatively the question was how
to extend Weierstrass' theory of elementary divisors from matrices to operators;
this was one of the initial motivations of 1. M. Gelfand and his collaborators for the
development of the theory of normed rings (Banach algebras). A closely related
problem is to study nonselfadjoint extensions of Hermitian operators. For instance,
one can consider a selfadjoint differential expression with nonselfadjoint bound
ary conditions; such nonselfadjoint boundary conditions (containing an arbitrary
complex number as a parameter) appear in the well-known book of the physicist
G. Gamow on the structure of atomic nucleus that was an important source of
motivation for M. S. Livsic.
Starting with von Neumann's theory of selfadjoint extensions of Hermitian oper
ators, M. S. Livsic has introduced the notion of the characteristic function of a non
selfadjoint extension of a Hermitian operator (with finite equal defficiency indices)
and used it to describe all such nonselfadjoint extensions.2 The ground breaking
paper of 1946 and the Doklady note of 1947 contain already the main ideas of the
theory of characteristic functions: the fact that the characteristic function deter
mines the corresponding operator essentially uniquely up to unitary equivalence
and the relation between factorizations of the characteristic function and invariant
subspaces of the operator. The general notion of the characteristic function of a
nonselfadjoint operator and its applications to the study of invariant subspaces and
the construction of canonical triangular models were developed by M. S. Livsic and
his collaborators (most notably M. S. Brodskii) during the following decade. An
important role was also played by the investigations of V. P. Potapov on the multi
plicative structure of J-contractive matrix valued functions which were motivated
by the work of M. S. Livsic. Following a suggestion of M. S. Livsic in the early
1960's, M. S. Brodskii introduced the notion of an operator colligation (or node)
which provides a natural and convenient framework for the study of nonselfadjoint
operators and the theory of characteristic functions.
The theory of characteristic functions is the cornerstone of the spectral theory
of nonselfadjoint and nonunitary operators; it allows to apply the entire theory of
bounded analytic functions to operator theoretic problems. Of equal and perhaps
even greater importance, operator theory leads to new problems in function theory
and provides new methods for a solution of old ones. The theory of characteristic
functions was recast in the language of functional models in the work of B. Sz.
Nagy-C. Foia§ and L. de Branges-J. Rovnyak in the 1960's.3
2These results form the second part of the second (doctoral) thesis of M. S. Livsic that he
defended at the Steklov Institute in Moscow in 1945; his opponents were S. Banach, 1. M. Gelfand,
M. A. Naimark, and A. 1. Plessner. The first part of this thesis is dedicated to the two dimensional
problem of moments; these results were never published. They are closely related to recent
investigations of M. Bakonyi, G. Naevdal and others.
3The original perspectives of these authors were quite different from those of M. S. Livsic and
his collaborators. The work of B. Sz.-Nagy and C. Foiru§ grew out of von Neumann's inequality
for contractions, Beurling's theorem on invariant subspaces of the shift operator, and Sz.-Nagy's
Editorial Introduction IX
A physical interpretation of a mathematical theory has always been for
M. S. Livsic an integral part of the theory itself. One would refer to him as a
"natural philosopher" in the English sense of the word encompassing the natural
sciences, mathematics and philosophy insofar as it relates to science and nature.
M. S. Livsic came to devote much of his scientific energy to the investigation
of a proper physical counterpart to the theory of nonselfadjoint operators and
their characteristic functions. Already in the middle of 1950's he showed that the
Heisenberg scattering matrix is a characteristic function of a certain nonselfadjoint
operator; these ideas were later formalized by P. D. Lax and R. S. Phillips in their
well-known approach to scattering theory, with numerous important applications
in mathematical physics. M. S. Livsic proceeded during the 1960's to introduce and
study open systems - in modern terminology, linear time-invariant input-output
dynamical systems4; the characteristic function of a nonselfadjoint operator (more
precisely, of an operator colligation) is the transfer function of the correspond
ing system. It took some time before the fundamental significance of this work
by M. S. Livsic was to be understood. Somewhat later in mathematical systems
theory, many similar results were obtained from a more algebraic perspective by
R. Kalman and his collaborators. As for operator theory and function theory, a
realization of a function as the transfer function of a linear system became an
essential tool during the last two decades. The article of V. Katsnelson in this
volume contains a historical survey of the theory of characteristic functions from
the point of view of realization theory.
In the early 1970's M. S. Livsic started considering the problem of linking system
theory and the theory of nonselfadjoint operators with Riemannian geometry. This
led naturally to developing a spectral theory for tuples of commuting nonselfadjoint
operators. The first major breakthrough here was achieved in a note published in
1978 in the Proceedings of the Georgian Academy of Sciences: M. S. Livsic proved
that a pair of commuting nonselfadjoint operators with finite nonhermitian ranks
satisfy an algebraic equation with constant coefficients. It seems then natural to
expect that the spectral theory of such a pair of operators leads to function theory
on an algebraic curve, the so called discriminant curve of the pair of operators
(rather than say to function theory of two independent complex variables as one
would assume a priori). In the early 1980's M. S. Livsic discovered that the proper
analogue of the notion of the characteristic function of a single nonselfadjoint op
erator is the so called joint characteristic function of a pair of operators, which
is a mapping of certain vector bundles (or more generally, of certain sheaves) on
the discriminant curve. The theory of commuting nonselfadjoint operators, their
discriminant curves and joint characteristic functions as well as the theory of cor
responding multidimensional systems were further developed by M. S. Livsic and
dilation theorem. The work of L. de Branges and J. Rovnyak evolved within the framework of
de Branges' theory of Hilbert spaces of entire functions.
4In the last chapter of his book on open systems published in 1966 (English translation in 1973),
M. S. Livilic considers also linear time-varying systems and suggests to study non-linear systems
as well; all of these became important topics of recent research.
x Editorial Introduction
his collaborators during the 1980's and 1990's. These theories combine the ideas of
operator theory, function theory and system theory with fundamental notions of
algebraic geometry such as line bundles and vector bundles on an algebraic curve,
Jacobian varieties and theta functions.
There can be little doubt that the notions of the discriminant curve and the
joint characteristic function will play an essential role in the decades to come as
the usual notion of the characteristic function did during the last fifty years. The
article of M. S. Livsic in this volume is dedicated to yet a further generalization
of these ideas, towards time-varying multidimensional systems and noncommuting
operators.
Beyond the obvious contributions of M. S. Livsic many in the mathematical
community have benefited from his generosity of thoughts and ideas. As an ex
ample, V. A. Marchenko recently shared with us that it was M. S. Livsic who
suggested to him in the early 1950's to consider the problem of recovering the
potential of a Sturm-Liouville operator from the scattering data.
Many of us who are well acquainted with the mathematical achievements of
M. S. Livsic are, however, unaware of the adverse circumstances in which he pur
sued his endeavors. After the hardships of the war and the evacuation he returned
to Odessa in 1945. And yet, still, in the postwar years M. G. Krein and the Odessa
school of functional analysis and operator theory became one of the most promi
nent victims of the new ethnic policies of the communist party and the zeal of
the local antisemites. M. S. Livsic moved to Kharkov in 1957. He remained in
Kharkov till 1975 when planning his immigration to Israel he moved to Tbilisi
as it was impossible to carry out from Kharkov. After immigrating to Israel in
1978, M. S. Livsic joined the Department of Mathematics and Computer Science
at Ben-Gurion University of the Negev in Beer-Sheva, where he is now Professor
Emeritus.
As we wish Moshe Livsic many more years of sharing his wonderful insights
with us, it seems appropriate to conclude this brief sketch of his mathematical
path up to the present and to introduce the articles that follow with the words of
Tyutchev:
HaM He ,n;aHo rrpe,n;yra,n;aTb,
KaK CJIOBO Harne oT30BeTC.lI, -
11 HaM COqYBcTBMe ,n;aeTC.lI,
KaK HaM ,n;aeTC.lI 6JIaro,n;aTL, ...
Daniel Alpay, Victor Vinnikov