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Bfm:978-3-0348-8543-0/1.Pdf 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, specificaUy the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use the permission of the copyright holder must be obtained. © 1994 Springer Basel AG Originally published by Birkhăuser Verlag in 1994 Softcover reprint of the hardcover 1st 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 . 0 • • • • • • • • • 54 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: Xm 111 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.
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