The theory of Toeplitz matrices and operators is a vital part of modern analysis, with applications to moment problems, orthogonal polynomials, approximation theory, integral equations, bounded- and vanishing-mean oscillations, and asymptotic methods for large structured determinants, among others. This friendly introduction to Toeplitz theory covers the classical spectral theory of Toeplitz forms and Wiener–Hopf integral operators and their manifestations throughout modern functional analysis. Numerous solved exercises illustrate the results of the main text and introduce subsidiary topics, including recent developments. Each chapter ends with a survey of the present state of the theory, making this a valuable work for the beginning graduate student and established researcher alike. With biographies of the principal creators of the theory and historical context also woven into the text, this book is a complete source on Toeplitz theory.
Author(s): Nikolski N.
Series: Cambridge studies in advanced mathematics 182
Publisher: Cambridge University Press
Year: 2020
Language: English
Pages: 454
Cover......Page 1
Front Matter
......Page 2
CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS......Page 3
Toeplitz Matrices and Operators......Page 5
Copyright
......Page 6
Dedication
......Page 7
Contents
......Page 9
Preface
......Page 15
Acknowledgments......Page 19
Biographies......Page 21
Figures
......Page 23
1 Why Toeplitz–Hankel? Motivations
and Panorama......Page 25
2 Hankel and Toeplitz: Sibling Operators
on the Space H2......Page 55
3 H2 Theory of Toeplitz Operators......Page 125
4 Applications: Riemann–Hilbert, Wiener–Hopf,
Singular Integral Operators (SIO)......Page 202
5 Toeplitz Matrices: Moments, Spectra,
Asymptotics......Page 254
Appendix A.
Key Notions of Banach Spaces......Page 353
Appendix B.
Key Notions of Hilbert Spaces......Page 357
Appendix C.
An Overview of Banach Algebras......Page 363
Appendix D.
Linear Operators......Page 372
Appendix E.
Fredholm Operators and the Noether Index......Page 383
Appendix F.
A Brief Overview of Hardy Spaces......Page 411
References......Page 419
Notation......Page 440
Index......Page 443
