Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

IDA function and asymptotic behavior of singular values of Hankel operators on weighted Bergman spaces

Zhijie Fan, Xiaofeng Wang, Zhicheng Zeng

Abstract

In this paper, we use the non-increasing rearrangement of ${\rm IDA}$ function with respect to a suitable measure to characterize the asymptotic behavior of the singular values sequence $\{s_n(H_f)\}_n$ of Hankel operators $H_f$ acting on a large class of weighted Bergman spaces, including standard Bergman spaces on the unit disc, standard Fock spaces and weighted Fock spaces. As a corollary, we show that the simultaneous asymptotic behavior of $\{s_n(H_f)\}$ and $\{s_n(H_{\bar{f}})\}$ can be characterized in terms of the asymptotic behavior of non-increasing rearrangement of mean oscillation function. Moreover, in the context of weighted Fock spaces, we demonstrate the Berger-Coburn phenomenon concerning the membership of Hankel operators in the weak Schatten $p$-class.

IDA function and asymptotic behavior of singular values of Hankel operators on weighted Bergman spaces

Abstract

In this paper, we use the non-increasing rearrangement of function with respect to a suitable measure to characterize the asymptotic behavior of the singular values sequence of Hankel operators acting on a large class of weighted Bergman spaces, including standard Bergman spaces on the unit disc, standard Fock spaces and weighted Fock spaces. As a corollary, we show that the simultaneous asymptotic behavior of and can be characterized in terms of the asymptotic behavior of non-increasing rearrangement of mean oscillation function. Moreover, in the context of weighted Fock spaces, we demonstrate the Berger-Coburn phenomenon concerning the membership of Hankel operators in the weak Schatten -class.

Paper Structure

This paper contains 17 sections, 25 theorems, 211 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Our results
  4. Difficulties and our strategy
  5. Organization of our paper
  6. Preliminaries
  7. Notation
  8. Singular values
  9. The class of weights $\mathcal{W}^\ast(\Omega)$
  10. Hankel operators and Toeplitz operators
  11. Technical Lemmas
  12. Boundedness and compactness characterizations
  13. Asymptotic behavior of singular values
  14. Singular values with slow decay
  15. Singular values with fast dacay
  16. ...and 2 more sections

Key Result

Theorem 1.4

Let $\omega \in \mathcal{W}^\ast(\Omega)$, $\delta\in (0,\delta _\omega)$ and $f\in\mathcal{S}\cap VDA_\omega(\Omega)$. We have the following conclusions:

Theorems & Definitions (63)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Definition 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 53 more