Topics in Operator Theory and Interpolation: Essays dedicated to M. S. Livsic on the occasion of his 70th birthday

240 Pages·1988·3.309 MB·English

by  Hari Bercovici, Ciprian Foias (auth.), I. Gohberg (eds.)

D
OT29
Operator Theory: Advances and Applications
Vol. 29
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)
BirkhauserV erlag
Basel· Boston· Berlin
lbpics in
Operator Theory
and Interpolation
s.
Essays dedicated to M. Livsic on
the occasion of his 70th birthday
Edited by
I. Gohberg
1988 Birkhauser Verlag
Basel· Boston· Berlin
Volume Editorial Office:
School of Mathematical Sciences
Tel Aviv University
Tel Aviv, Israel
CIP-Titelaufnahme der Deutschen Bibliothek
Topics in operator theory and interpolation : essays dedicated
to M. S. Livsic on the occasion of his 70th birthday / ed. by I.
Gohberg. - Basel ; Boston ; Berlin : Birkhauser, 1988
(Operator theory ; Vol. 29)
NE: Gochberg, Izrai!' [Hrsg.]; Livsic, M. S.: Festschrift; 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 ,>YerwertungsgesellschaftWort«, Munich.
© 1988 Birkhauser Verlag Basel
Softcover reprint of the hardcover 1st edition 1988
ISBN-13: 978-3-7643-1960-1 e-ISBN-13: 978-3-0348-9162-2
DOl: 10.1007/978-3-0348-9162-2
CONTENTS
Biography of M.S. Livsic 7
List of publications
of ~LS. Livsic 17
BERCOVICI, H. On skew Toepl itz
FOIAS, C. Operators, I. 21
TANNENBAUM, A.
CAREY, R. W., On local index and the
PINCUS, J.E. cocycle property for
Lefschetz numbers 45
DAVIS, C. Completing a matrix so
as to minimize the rank 87
DEWILDE, P. The generalized Schur
DEPRETTERE, E.F.S. algorithm: Approximation
and hierarchy .... 97
DYM, H. A new class of contractive
GOHBERG, I. interpolants and maximum
entropy principles 117
FEINTUCH, A. Distance formulas for
FRANCIS, B. operator algebras arising
in optimal control problems 151
FO I AS, C. On the Schur representation
FRAZHO, A.E. in the commutant lifting
theorem II 171
GOHBERG, I. Nodes and realizations of
KAA S HOE K, ~: . A. rational matrix functions:
LERER, L. Minimality theory and
appl ications 181
KERCHY, L. On the multiplicity of the
commutant of operators 233
M. S. LlVSIC
7
BIOGRAPHY OF M.S. LIVSIC
Moshe Livsic (Mikhail Samuilovich livsic) was born on the
4th of July. 1917 in the small town of Pokotilova near Uman. a
province of Kiev. in the Ukraine (according to the 1897 census,
Pokotilova numbered 3030 citizens, including 1670 Jews). When he
was four years old. his family moved to Odessa where his father
held the position of associate professor of mathematics in an
academic institute. His father's influence on Moshe, an only
child. was very great, and until today he has fond memories of
the Yiddish songs and Jewish prayers sung by his father, who
before his mathematical career, was a cantor at the synagogue.
Moshe livsic's father often spoke about the great mathemati-
cians who were active at that time in Odessa: N.G. Chebataryov,
an outstanding algebraist; V.F. Kagan, an outstanding expert in
geometry, especially non-Euclidean; Yu. I. Timchenko, an expert
in mechanics; M.G. Krein, who was at that time a young post-
graduate student; and s.o. Shatunovsky. They were all close
friends of Moshe's father.
S. o. Shatunovsky was an especially intimate friend of the
elder livsic. Shatunovsky was an excellent lecturer and an
extraordinary person, well known to many of the townsfolk. Even
today the older generation still recall the professor whose
fascinating lectures in mathematics were attended by hundreds of
students from all the faculties of the university.
In the West relatively little is known about the work of s.o.
Shatunovsky (1859-1929). He was one of the first representa
tives of constructive mathematics, and a pioneer in intuitionist
logic and modern algebra. He devoted much time to the law of the
excluded third. and in 1901 he was the first to indicate that the
formal transfer of this law to infinite sets is not obvious. In
8
his dissertation, completed in 1917, S.D. Shatunovsky constructed
a foundation of algebra and in particular of Galois theory as a
theory of congruences with respect to the functional moduli
introduced by Cauchy. without invoking the law of the excluded
third to infinite sets. S.D. Shatunovsky found an original
generalization of the limit notion. Without using limits. he
defined the volume of a polyhedron as a certain invariant.
Independently of Hilbert. and approximately at the same time
(1897-1898) he laid the axiomatic foundation of the theory of
areas. In 1910, together with V.F. Kagan. S.D. ShatunovskY
founded a mathematical publishing house in Odessa. This publish
hing house. "Matesis". played a significant role in mathematical
education and the popularization of mathematics in Russia.
"Matesis" published translations of European classics in
mathematics and a number of textbooks. including the well-known
book of Dedekind on the theory of irrational numbers. In 1923
"Matesis" published S.D. Shatunovsky's book "Introduction to
Analysis" which contained his lectures on the subject. "Matesis"
continued operating until some time after the October revolution.
In 1931 Moshe graduated from school. which at that time
comprised seven grades. At school he became friendly with Israel
M. Glazman. and their friendship continued intermittently until
the tragic death of this outstanding mathematician on May 30th.
1968. At the age of sixteen. the two friends worked out a plan
for their future education which culminated in a deep study of
philosophy (Kant, Hegel. and the works of a number of English and
French philosophers). It was clear to them that the study of
philosophy in the 20th century had no value without a fundamental
knowledge of the natural sciences and so they planned to study
physics and chemistry first. However. since a study of the
natural sciences was impossible without a fundamental knowledge
of mathematics. the two friends reached the logically inevitable
conclusion that they should begin by studying mathematics. Many
years later M.S. Livsic light heartedly summed up the youngsters'
program: "I succeeded in thoroughly studying some fields in
9
mathematics. I succeeded less in studying some fields in physics.
There was no time for philosophy." Nevertheless a tendency to a
philosophical understanding of scientific results remained.
As a youth Moshe Livsic was attracted to the new technology
of radio, and he dreamed of becoming a radio technician. After
graduating from school he entered the radio class at the Techni
cal College for Communication in Odessa. However, in 1933, he
interrupted these studies to enroll in the newly created Depart
ment of Physics and Mathematics at the Odessa State University.
During the first years at the university mathematics was
taught by M.G. Krein. F.R. Gantmakher (who shortly thereafter
left for Moscow). and M. A. Naimark. who was a graduate student.
assistant and. subsequently. a colleague of M.G. Krein. Later.
when Moshe Livsic was already a third year student, B. Ya. Levin,
a prominent specialist in the theory of analytic functions.
joined the faculty.
Those who influenced Moshe Livsic most during his university
years were M. G. Krein and B. Ya. Levin. He especially remembers
B. Ya. Levin's course on theory of functions of a complex
variable. and M. G. Krein's courses on integral equations and on
the differential equations of mathematical physics, both from
the viewpoint of functional analysis. These courses were on the
frontiers of research of that time.
Among his fellow students and friends, Moshe Livsic remem-
bers especially his schoolfriend I.M. Glazman. He also remembers
V. Smushkovich, a very talented mathematician and a handsome
young man (lost in the Second World War), and A.P. Artyomenko,
the most talented student of M.G. Krein (and much older than the
others). Artyomenko obtained important results in the theory of
Hermitian positive-definite functions. He and M.S. Livsic had a
warm relationship. After World War II. A.P. Artyomenko
disappeared and all attempts to find him were of no avail. A
younger student who entered the university a year after Moshe
Livsic was V.P. Potapov, who later also became an eminent mathe-