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Besov spaces in operator theory

V. V. Peller

Abstract

The survey is devoted to diverse applications of Besov classes in operator theory. It is illustrated how Besov classes are used to describe Hankel operators of Schatten--von Neumann classes; various applications of this description are considered. Next, we discuss the role of Besov classes in norm estimates of polynomials of power bounded operators on Hilbert space and related estimates of Hankel matrices in tensor products of the spaces $\ell^1$ and $\ell^\infty$. An essential part of the survey is devoted to the role of Besov spaces in various problems of perturbation theory when studying the behavior of functions of a single operator or of a collection of operators under their perturbation.

Besov spaces in operator theory

Abstract

The survey is devoted to diverse applications of Besov classes in operator theory. It is illustrated how Besov classes are used to describe Hankel operators of Schatten--von Neumann classes; various applications of this description are considered. Next, we discuss the role of Besov classes in norm estimates of polynomials of power bounded operators on Hilbert space and related estimates of Hankel matrices in tensor products of the spaces and . An essential part of the survey is devoted to the role of Besov spaces in various problems of perturbation theory when studying the behavior of functions of a single operator or of a collection of operators under their perturbation.
Paper Structure (30 sections, 244 equations)

This paper contains 30 sections, 244 equations.

Table of Contents

  1. Introduction
  2. Definitions of Besov classes
  3. Besov classes on the unit circle
  4. Besov classes on Euclidean spaces
  5. Inhomogeneous Besov classes on Euclidean spaces.
  6. An introduction to Schatten--von Neumann classes
  7. Hankel operators and Besov classes
  8. Classical Hankel operators and Hankel matrices
  9. Generalized Hankel matrices
  10. Certain applications of the criterion of the membership of Hankel operators in classes $\boldsymbol{{\boldsymbol S}_p}$
  11. A characterization of Besov classes in terms of rational approximation
  12. Commutators of multiplication operators and the Riesz projection
  13. Hankel operators as integral operators on $\boldsymbol{L^2[0,\infty)}$; commutators of multi-plication operators and the Hilbert transform
  14. Wiener--Hopf operators on a finite interval
  15. The I.A. Ibragimov--V.N. Solev problem on stationary random processes
  16. ...and 15 more sections