Toeplitz Operators and Related Topics: The Harold Widom Anniversary Volume Workshop on Toeplitz and Wiener-Hopf Operators, Santa Cruz, California, September 20–22,1992

207 Pages·1994·4.88 MB·English

by  Harold Widom (auth.)

Operator Theory
Advances and Applications
Vol. 71
Editor
I. Gohberg
Editorial Office: T. Kailath (Stanford)
School of Mathematical H.G. Kaper (Argonne)
Sciences S.T. 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 (Blackburg) M. Rosenblum (Charlottesville)
A. Ben-Artzi (Tel Aviv) J. Rovnyak (Charlottesville)
H. Bercovici (Bloomington) D.E. Sarason (Berkeley)
A. Bottcher (Chemnitz) H. Upmeier (Lawrence)
L. de Branges (West Lafayette) S.M. Verduyn-Lunel (Amsterdam)
K. Clancey (Athens, USA) D. Voiculescu (Berkeley)
L.A. Coburn (Buffalo) H. Widom (Santa Cruz)
K.R. Davidson (Waterloo, Ontario) D. Xia (Nashville)
R.G. Douglas (Stony Brook) D. Yafaev (Rennes)
H. Dym (Rehovot)
A. Dynin (Columbus)
P.A. Fillmore (Halifax) Honorary and Advisory
C. Foias (Bloomington) Editorial Board:
P.A. Fuhrmann (Beer Sheva) P.R. Halmos (Santa Clara)
S. Goldberg (College Park) T. Kato (Berkeley)
B. Gramsch (Mainz) P.D. Lax (New York)
G. Heinig (Chemnitz) M.S. Livsic (Beer Sheva)
J.A. Helton (La Jolla) R. Phillips (Stanford)
M.A. Kaashoek (Amsterdam) B. Sz.-Nagy (Szeged)
Toeplitz Operators
and Related Topics
The Harold Widom Anniversary Volume
Workshop on Toeplitz and Wiener-Hopf Operators,
Santa Cruz, California, September 20-22,1992
Edited by
E.L. Basor
1. Gohberg
Springer Basel AG
Volume Editorial Office:
Raymond and Beverly Sackler Facuity of Exact Sciences
School of Mathematical Sciences
Tel Aviv University
69978 Tel Aviv
Israel
A CIP catalogue record for this book is available from the Library ofCongress, Washington D.C., USA
Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Toeplitz operators and related topics : the Harold Widom
anniversary volume; workshop on Toeplitz and Wiener-Hopf
Operators, Santa Cruz, California, September 20-22, 1992 /
[voI. ed. office: Raymond and Beverly Sackler Faculty of Exact
Sciences, School of Mathematical Sciences, Tel Aviv
University]. Ed. by E.L. Basor; 1. Gohberg. - Basel ; Boston;
Berlin: Birkhăuser, 1994
(Operator theory; VoI. 71)
ISBN 978-3-0348-9672-6 ISRN 978-3-0348-8543-0 (eRook)
DOI 10.1007/978-3-0348-8543-0
NE: Basor, Estelle L. [Hrsg.]; Workshop on Toeplitz and Wiener-Hopf
Operators <1992, Santa Cruz, Calif.>; Bet has-Sefer 1e-Madda'e ham
Matematîqa <Ramat- A vÎv> / haf-FaqUlta le-Madda'Îm
MeduyyaqÎm'al Sem Raymond u-Beverly Saqler; GT
This work is subject to copyright. AU rights are reserved, whether the whole or part of the material is concerned,
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must be obtained.
© 1994 Springer Basel AG
Originally published by Birkhăuser Verlag in 1994
Softcover reprint of the hardcover 1s t edition 1994
Camera-ready copy prepared by the editors
Printed on acid-free paper produced from chlorine-free pulp
Cover design: Heinz Hiltbrunner, Basel
ISBN 978-3-0348-9672-6
987654321
Table of Contents
E.L. BasoT, E.M. Landesman
Biography of H. Widom ........................ IX
1. Kaplansky
To Harold Widom on his 60th birthday . Xll
Harold Widom's Publications . xv
H. Widom
Eigenvalue distribution for nonselfadjoint Toeplitz matrices . 1
H. Widom
Random Hermitian matrices and (nonrandom) Toeplitz matrices . . . . . 9
E.L. Basor, K.E. Morrison
The extended Fisher-Hartwig conjecture for symbols with multiple
jump discontinuities .......... 16
1. Introduction . . . . . . . . . . . . 16
2. Localization theory for -1 < Re{3r < 0 18
3. Some special cases 24
References . . . . . . . . . . . 28
H. Bercovici, C. Foias, A. Tannenbaum
A relative Toeplitz-Hausdorff theorem . . . . . . . . . . . . . . . . . . 29
A. Bottcher, B. Silbermann
Operator-valued Szego-Widom limit theorems 33
1. Introduction . . . . . . . . . . . . . . . 33
2. Toeplitz and Hankel operators with operator-valued symbols 34
3. Finite section method . . . . . . . . 36
4. The first Szego-Widom limit theorem . 38
5. The strong Szego-Widom limit theorem 44
6. Continuous symbols. 49
References . . . . . . . . . . . . . . 52
M. Cotlar, C. Sadosky
The Adamjan-Arov-Krein theorem in general and regular
representations of R2 and the symplectic plane . . . . . 54
Introduction . . . . . . . . . . . . 54
0 • • • • • • • • •
1.Hankel forms in discrete and continuous evolution systems . 56
2. The A-A-K theorem in general and regular representations of Ii! 61
3. The A-A-K theorem for general representations of R2 and (R2, [ , ]) 63
4. A-A-K theorems in the regular representation of R2 ..... 68
5. The A-A-K theorem in the regular representation of the symplectic plane 71
References . . . . . 77
J. Gohberg, M.A. Kaashoek
Projection method for block Toeplitz operators with
operator-valued symbols . . . . . . . 79
O. Introduction . . . . . . . . . . . . . . . . . . 79
1. Projection method for compressions. . . . . . . . 81
2. Applications of block Toeplitz operators with operator-valued symbols 87
3. Projection method with two directions 91
4. The projection method for block pair operators . 95
5. The theorem for the non-Toeplitz case 100
References . . . . . . . . . . . . . . 104
J. Gohberg, N. Krupnik
Szego-Widom-type limit theorems 105
1. Introduction . . . . . . . . . 105
2. The main results . . . . . . . 108
3. Limit theorems for the algebra K:Z: 111
Xm
4. Illustrative examples 115
5. Proofs of theorems 2.1, 2.2 . 118
References . . . . . 120
Y. Karlovich, J. Spitkovsky
(Semi)-Fredholmness of convolution operators on the spaces
of Bessel potentials 122
1. Introduction . . 122
2. Main results . . 125
3. Auxiliary results 130
4. Operators W K,w 139
5. Invertibility and factorability of local representatives 145
References . . . . . . . . . . . . . . . . . . . . 150
D. Samson
Kernels of Toeplitz operators 153
1. Introduction . . . . . . 153
2. De Branges-Rovnyak spaces 155
3. Hitt's theorem . . 158
4. Hayashi's theorem 160
5. Rigid functions 163
References . . 163
S. Treil, A. Volberg
A fixed point approach to Nehari's problem and its applications 165
1. Abstract Nehari's theorem. . . . . . . . . . . . . 165
2. Main examples and corollaries . . . . . . . . . . . 167
3. Iokhvidov-Ky Fan theorem and some of its applications:
A generalized AAK theorem . . . . . . 172
4. Proof of the generalized Nehari's theorem 176
5. Proof of the Iokhvidov-Ky Fan theorem 182
References . . . . . . . . . . . . . . . 185
HAROLD WIDOM
Operator Theory: IX
Advances and Applications, Vol. 71
© 1994 Birkhiiuser Verlag Basel/Switzerland
BIOGRAPHY OF H. WIOOM
Estelle L. Basor and Edward M. Landesman
A special meeting on Toeplitz and Wiener-Hopf operators was held in Santa Cruz,
California on September 20-22, 1992, to celebrate the 60th birthday of Harold Widom and to
recognize his contributions to the mathematical community. The meeting was sponsored jointly by
the Mathematics Department of the University of California at Santa Cruz and the Mathematical
Sciences Research Institute in Berkeley. The authors of this essay both spoke about the life and
work of Harold Widom at the conference, and we are repeating many of our remarks for the
present readers.
The career of Harold Widom has been distinguished by his research, teaching, and service
to the mathematical profession. He was born September 23, 1932, in Newark, New Jersey
during the heart of the Depression. His parents were born in eastern Europe, his father in
Kamentz-Podolskii in the Ukraine and his mother in Mielic, Poland. Both of his parents came to
the United States in 1914, when his mother was 15 years old and his father 22. They met in New
York and were married there in 1924.
Harold grew up in Brooklyn, New York. He went to Stuyvesant High School where he
was captain of the math team. Coincidentally, the captain of the rival team at the Bronx High
School of Science was Henry Landau, who also attended the conference and has been a long-time
friend and colleague of Harold's. Also attending the conference was Harold's brother Benjamin,
who is five years older and is Professor of Chemistry at Cornell University.
After high school, he attended the City College of New York and the University of
Chicago, where at the age of twenty he received a Master's Degree. Three years later, in 1955, he
obtained his Ph.D. in mathematics, having written a dissertation in functional analysis, under the
direction of Irving Kaplansky. He began his academic career as an instructor at Cornell University
where he rose through the ranks to become full professor in 1965. At Cornell, his interests
focused on classical analysis and operator theory and he proved many of the early beautiful
theorems about Toeplitz operators. In particular, he gave necessary and sufficient conditions for a
Toeplitz operator to be invertible, proved an index theorem for Toeplitz operators with piecewise
continuous symbol, and proved the still somewhat mysterious result that the spectrum of a Toeplitz
operator is connected. This analysis was characterized by a special style --very deep theorems
with difficult proofs, motivated and described in a most elegant, clear way.
x
E. L. Basor and E. M. Landesman
In 1968, after visiting Stanford University for one year, a rumor surfaced that Harold
enjoyed California weather. Happily for the University of California at Santa Cruz the
mathematics department was able to convince him to join its faculty in the fall of 1968. At Santa
Cruz, his research evolved from the study of Szego Limit Theorems. This theorem gave the two
first terms in the asymptotic expansion of the determinants of finite Toeplitz matrices. The proofs
of the Szego theorems were difficult, indirect, and the value of the constant term did not follow
intuitively from anything else.
Harold changed this state of affairs in 1976 when he gave a beautiful operator theory proof
of the strong Szego Limit Theorem, showed that the constant term was simply an infinite
determinant, and extended the result to the block matrix case. He also extended the original Szego
result to Wiener-Hopf operators, Toeplitz operators with singular symbols, and, in the most
general setting, to pseudodifferentail operators.
For pseudodifferential operators on manifolds he developed a complete symbolic calculus,
which allowed him to find a complete Szego expansion for n-dimensional convolution operators on
general domains. He also showed how the Szego expansion and analogues of the heat expansion
of the Laplace operator could be computed with one common pseudodifferential operator approach.
The is the subject of his book Asymptotic Expansions for Pseudodifferential Operators on
Bounded Domains, which appeared in Lecture Notes in Mathematics. Most recently, his interests
have turned to the study of random matrices and their relationships to Toeplitz and Wiener-Hopf
operators.
Throughout his career, Harold has been known as a superb and inspiring teacher. The
same "Widom" style so evident in his research is evident also in his teaching. His lectures are
polished, clear, and beautifully motivated. His ability to uncover the heart of a subject and to make
complicated details seem simple has attracted many students to analysis. He has directed the
dissertations of seven Ph.D. students. His first student, Lydia Luquet, was a student at Cornell.
The six others include the first author, Ray Roccaforte, Xiang Fu, Shuxian Lou, Richard Libby,
and Bobette Thorsen.
On a personal note, the first author was a student in an undergraduate real analysis course
taught by Harold in the fall and winter of 1968-69 at Santa Cruz. For her, those courses were the
most delightful in her undergraduate career. Harold managed to teach analysis in a very rigorous
fashion, yet allowed the intuition and beauty to come through. His love and respect for
mathematics is always apparent and has influenced many a student and colleague.
Harold has also worked in many ways for the mathematics community. He has been on
the editorial boards for the Journal of Integral Equations and Operator Theory and the journal for
Asymptotic Analysis and an organizer for several conferences at Oberwolfach. In addition to
Asymptotic Expansions for Pseudodifferential Operators on Bounded Domains, he has written
two books in the Van Nostrand Mathematical Studies series, Lectures on Integral Equations and
Lectures on Measure and Integration.