Toeplitz Matrices and Singular Integral Equations: The Bernd Silbermann Anniversary Volume

326 Pages·2002·9.14 MB·English

by  Albrecht Böttcher (auth.)

Operator Theory: Advances and
Applications
VoI. 135
Editor:
1. Gohberg
Editorial Oftice:
School of Mathematical
Sciences
Tel Aviv University
Ramat Aviv, Israel
Ed itorial Board: P. Lancaster (Calgary)
J. Arazy (Haifa) L.E. Lerer (Haifa)
A. Atzmon (Tel Aviv) B. Mityagin (Columbus)
J. A. BaII (Blacksburg) V. V. Peller (Manhattan, Kansas)
A. Ben-Artzi (Tel Aviv) J. D. Pincus (Stony Brook)
H. Bercovici (Bloomington) M. Rosenblum (Charlottesville)
A. Battcher (Chemnitz) J. Rovnyak (Charlottesville)
K. Clancey (Athens, USA) D. E. Sarason (Berkeley)
L. A. Coburn (Buftalo) H. Upmeier (Marburg)
K. R. Davidson (Waterloo, Ontario) S. M. Verduyn Lunei (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 (Ren nes)
P. A. Fuhrmann (Beer Sheva)
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)
Toeplitz Matrices and
Singular Integral Equations
The Bernd Silbermann Anniversary Volume
Albrecht Bottcher
Israel Gohberg
Peter Junghanns
Editors
Springer Basel AG
Editors:
Albrecht Btittcher Peler Junghanns
Faculty of Mathematics Faculty of Mathematics
Technical University Chemnitz Technical University Chemnitz
09107 Chemnitz 09107 Chemnitz
Germany Germany
e-mail: [email protected] e-mail: [email protected]
Israel Gohberg
School of Mathematical Sciences
Raymond and Beverly Sack1er
Faculty of Exact Sciences
Tel Aviv University
Ramat Aviv 69978
Israel
e-mail: [email protected]
2000 Mathematics Subject C!assification 47-06; 47A56, 47B35, 47B48, 47N20, 47N70
A CIP catalogue record for this book is available from the
Library of Congress, Washington D.C., USA
Deutsche Bibliothek Cataloging-in-Publication Data
Toeplitz matrices and singular integral equations : the Bemd Silbermann
anniversary volume / Albrecht Btittcher ... ed. - Basel ; Boston; Berlin:
Birkhliuser, 2002
(Operator theory ; VoI. 135)
ISBN 978-3-0348-9471-5 ISBN 978-3-0348-8199-9 (eBook)
DOI 10.1007/978-3-0348-8199-9
This work is subject to copyright. AII rights are reserved, whether the whole or part of the material is concemed,
specifically the rights of translation, reprinting, re-use of iIIustrations, recitation, broadcasting, reproduction on microfi1~
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© 2002 Springer Basel AG
Originally published by Birkhăuser Verlag Basel, Switzerland in 2002
Printed on acid-free paper produced from chlorine-free pulp. TCF
00
Cover design: Heinz Hiltbrunner, Basel
www.birkhauser-science.com
987654321
Contents
Editorial Preface .......................................................... vii
Portrait of Bernd Silbermann ............................................. viii
A. Bottcher
Essay on Bernd Silbermann .......................................... 1
Publications of Bernd Silbermann ......................................... 13
J.A. Ball, L. Rodman and I.M. Spitkovsky
Toeplitz Corona Problem for Algebras of
Almost Periodic Functions ........................................... 25
H. Bart, T. Ehrhardt and B. Silbermann
Sums of Idempotents in the Banach Algebra
Generated by the Compact Operators and the Identity ............... 39
E.L. Basor and T. Ehrhardt
Asymptotic Formulas for the Determinants of
Symmetric Toeplitz plus Hankel Matrices............................ 61
A. Bottcher
On the Determinant Formulas by Borodin, Okounkov,
Baik, Deift and Rains................................................ 91
A. Bottcher, S. Grudsky and A . Kozak
On the Distance of a Large Toeplitz Band Matrix
to the Nearest Singular Matrix ...................................... 101
L.P. Castro, R. Duduchava and F.-G. Speck
Singular Integral Equations on Piecewise Smooth Curves
in Spaces of Smooth Functions ....................................... 107
V.D. Didenko and B. Silbermann
Spline Approximation Methods for the Biharmonic
Dirichlet Problem on Non-smooth Domains.......................... 145
B. Fritzsche, B. Kirstein and A. Lasarow
On Rank Invariance of Schwarz-Pick-Potapov Block Matrices
of Matrix-valued CaratModory Functions............................ 161
I. Gohberg, M.A. Kaashoek and F. van Schagen
Finite Section Method for Linear Ordinary
Differential Equations Revisited ..................................... 183
VI Contents
G. Heinig and K. Rost
Fast Algorithms for Skewsymmetric Toeplitz Matrices ............... 193
P. Junghanns, K. Muller and K. Rost
On Collocation Methods for Nonlinear Cauchy Singular
Integral Equations ................................................... 209
Yu. Karlovich and B. Silbermann
Local method for nonlocal operators on Banach spaces 235
G. Mastroianni and G. M onegato
Numerical Solution of Mellin Type Equations
via Wiener-Hopf Equations.......................................... 249
A. Pietsch
A 1-parameter Scale of Closed Ideals Formed
by Strictly Singular Operators ....................................... 261
V.S. Rabinovich and S. Roch
Local Theory of the Fredholmness of Band-dominated
Operators with Slowly Oscillating Coefficients ....................... 267
S. Serra-Capizzano
More Inequalities and Asymptotics for Matrix Valued
Linear Positive Operators: the Noncommutative Case 293
H. Widom
Toeplitz Determinants, Random Matrices
and Random Permutations .......................................... 317
Editorial Preface
This volume is dedicated to Bernd Silbermann on the oc
casion of his sixtieth birthday. It consists of selected papers
devoted to the inexhaustible and ever-young fields of Toeplitz
matrices and singular integral equations, and thus to areas
Bernd Silbermann has been enriching by fundamental con
tributions for the last three decades.
Most authors of this volume participated in the conference
organized and sponsored by the Department of Mathematics
(under the deans Dieter Happel and Jiirgen vom Scheidt) of
the Chemnitz University of Technology in honor of Bernd
Silbermann in Pobershau, April 8-12, 2001. The majority of
the papers presented here are based on the talks given on that
conference. We thank all contributors for their enthusiasm
when preparing the articles for this volume.
BERND SILBERMANN
Operator Theory:
Advances and Applications, Vol. 135, 1-12
© 2002 Birkhiiuser Verlag Basel/Switzerland
Essay on Bernd Silbermann
Albrecht Bottcher
Bernd Silbermann was born on 6 April 1941 in Langhennersdorf, a village in Sax
ony. His parents were farmers. To this day, he is proud of his ability to drive a
tractor. He went to school in Langhennersdorffrom 1947 to 1955, and in the sub
sequent two years he apprenticed to a grocer (and really sold fish). From 1958 to
1962 he attended a school in Chemnitz, and from 1962 to 1967, in the heyday of
Soviet mathematics, he was a student of mathematics at the Lomonosov University
in Moscow. His lecturers included such eminent mathematicians as P.S. Alexan
drov, A.G. Kurosh, and G.E. Shilov. His diploma paper was supervised by E.A.
Gorin and A.Ya. Helemskii and was devoted to the structure of radicals in certain
normed rings. Since 1966 he has been married to Ludmilla Pavlovna; their children
Sergej and Katja were born in 1971 and 1974. In 1967, Silber mann moved back to
Chemnitz and started working under the supervision of Siegfried Prossdorf.
1967-1969. First beat of the drum
Following the advice of Prossdorf, he embarked on singular integral, Toeplitz and
Wiener-Hopf operators whose symbols have zeros. From the work of Chebotarev,
Cherskii, Dybin, Haikin, Karapetyants, and Prossdorf, to mention only a few prin
cipal figures, it was known that if the symbol is in Coo and has only a finite number
of zeros of integral orders, then the corresponding Toeplitz operator is not normally
solvable on the usual Banach spaces, but it is Fredholm on certain Frechet spaces
of test functions and distributions. The big question of those days was whether
the converse is true. Silbermann tackled this question and showed by an ingenious
proof that the answer is yes: if a Toeplitz operator with a Coo symbol is Fredholm
on the spaces of test functions or distributions mentioned, then the symbol has
at most a finite number of zeros of integral orders. This result was a beat of the
drum. It advanced him immediately to the first row of researchers in the field and
it is up to the present the brightest star in the vault of Toeplitz operators whose
symbols have zeros. The result is proved in Silbermann's paper "On singular inte
gral operators in spaces of infinitely differentiable and generalized functions" (in
Russian), which appeared in the then famous journal "Matematicheskie Issledova
nia" in 1971. This paper is Silbermann's actual opus 1 (another paper appeared
earlier but was written later), and it has become one of his most frequently cited
papers.
2 A. Bottcher
1970-1977. The years with Prossdorf
Within a short time, Silbermann grew from a student of Prossdorf's to a co
worker of equal rank. Together they made numerous significant contributions to
the theory of singular integral operators that are not normally solvable, and to
projection methods for such operators. The other co-workers of Silbermann during
that period were Uwe Kohler, Christian Meyer, Johannes Steinmiiller, Karla Rost,
and Johannes Elschner. In 1970, Silbermann defended his Dr. rer. nat. thesis, and
only four years later, in 1974, he completed his habilitation thesis. He was then 33
years old.
One cannot remember those years without mentioning Israel Gohberg. He
visited Chemnitz in the late 60s and it was on his suggestion that Prossdorf and
Silbermann started working on projection methods for singular integral operators
whose symbols have zeros. Moreover, in 1974, Georg Heinig came to Chemnitz. He
had then just accomplished his PhD under Gohberg's supervision and brought a
good deal of Gohberg's spirit to Chemnitz.
In 1975, Prossdorf left Chemnitz and went to Berlin. The large amount of
mathematics developed by Prossdorf and Silbermann during those years resulted in
their book "Projektionsverfahren und die naherungsweise Losung singularer Glei
chungen", which appeared in 1977. In 1976, Silbermann was appointed professor
(Dozent) in Chemnitz. He then had one book and more than 20 published papers.
In those times, this was an outstanding balance.
1978-1979. I entered the scene
Now it is time to introduce myself. I became a mathematics student in Chemnitz in
1975. I attended Silbermann's lecture course Analysis I-III and was fascinated by
his charismatic teaching ability. In 1977, a couple of weeks before Easter, I turned
to him with the request for a research problem. I remember perfectly telling him
that I intended to bridge the time until Easter by doing some research. He gave
me a problem and added on his turn that he would be impressed if I solved it by
Christmas. The problem was as follows. Prossdorf had shown that if a function on
the unit circle is Holder continuous with the exponent a, then the partial sums
of the Fourier series converge to the function in the ,8th Holder norm provided
,8 < a. Silbermann asked me to check whether such a result is also true locally,
that is, whether the partial sums of the Fourier series of a function that is Holder
continuous with the exponent a on some arc I converge in the ,8th Holder norm
(,8 < a) to the function on every arc J properly contained in I. I solved the
problem (in the negative), but this was in 1978, many months after the Christmas
of 1977.
Some time in the second half of the 70s, Silbermann learned of the formula
(Pap)-l P = Pa-1P _ Pa-1Q(Qa-1Q)-lQa-1P
from Anatoli Kozak of Rostov-on-Don. He fell in ardent love with this formula and
was soon able to do real wonders with it. One of those miracles was a new proof