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Operators, Functions, and Systems: An Easy Reading: Volume 1: Hardy, Hankel, and Toeplitz
Nikolai K. Nikolski, University of Bordeaux I, Talence, France
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Mathematical Surveys and Monographs
2002; 461 pp; softcover
Volume: 92
ISBN-10: 0-8218-4933-6
ISBN-13: 978-0-8218-4933-0
List Price: US$113
Member Price: US$90.40
Order Code: SURV/92.S
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Together with the companion volume by the same author, Operators, Functions, and Systems: An Easy Reading. Volume 2: Model Operators and Systems, Mathematical Surveys and Monographs, Vol. 93, AMS, 2002, this unique work combines four major topics of modern analysis and its applications:

A. Hardy classes of holomorphic functions,

B. Spectral theory of Hankel and Toeplitz operators,

C. Function models for linear operators and free interpolations, and

D. Infinite-dimensional system theory and signal processing.

This volume contains Parts A and B.

Hardy classes of holomorphic functions is known to be the most powerful tool in complex analysis for a variety of applications, starting with Fourier series, through the Riemann \(\zeta\)-function, all the way to Wiener's theory of signal processing.

Spectral theory of Hankel and Toeplitz operators becomes the supporting pillar for a large part of harmonic and complex analysis and for many of their applications. In this book, moment problems, Nevanlinna-Pick and Carathéodory interpolation, and the best rational approximations are considered to illustrate the power of Hankel and Toeplitz operators.

The book is geared toward a wide audience of readers, from graduate students to professional mathematicians, interested in operator theory and functions of a complex variable. The two volumes develop an elementary approach while retaining an expert level that can be applied in advanced analysis and selected applications.

Readership

Graduate students and research mathematicians interested in analysis.

Table of Contents

An invitation to Hardy classes/Contents
  • Foreword to Part A
  • Invariant subspaces of \(L^2(\mu)\)
  • First applications
  • \(H^p\) classes. Canonical factorization
  • Szegö infimum, and generalized Phragmén-Lindelöf principle
  • Harmonic analysis in \(L^2(\mathbb{T},\mu)\)
  • Transfer to the half-plane
  • Time-invariant filtering
  • Distance formulae and zeros of the Riemann \(\zeta\)-function
Hankel and Toeplitz operators/Contents
  • Foreword to Part B
  • Hankel operators and their symbols
  • Compact Hankel operators
  • Applications to Nevanlinna-Pick interpolation
  • Essential spectrum. The first step: Elements of Toeplitz operators
  • Essential spectrum. The second step: The Hilbert matrix and other Hankel operators
  • Hankel and Toeplitz operators associated with moment problems
  • Singular numbers of Hankel operators
  • Trace class Hankel operators
  • Inverse spectral problems, stochastic processes and one-sided invertibility
  • Bibliography
  • Author index
  • Subject index
  • Symbol index
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