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IDA and Hankel operators on Fock spaces

Zhangjian Hu, Jani A. Virtanen

Abstract

We introduce a new space IDA of locally integrable functions whose integral distance to holomorphic functions is finite, and use it to completely characterize boundedness and compactness of Hankel operators on weighted Fock spaces. As an application, for bounded symbols, we show that the Hankel operator $H_f$ is compact if and only if $H_{\bar f}$ is compact, which complements the classical compactness result of Berger and Coburn. Motivated by recent work of Bauer, Coburn, and Hagger, we also apply our results to the Berezin-Toeplitz quantization.

IDA and Hankel operators on Fock spaces

Abstract

We introduce a new space IDA of locally integrable functions whose integral distance to holomorphic functions is finite, and use it to completely characterize boundedness and compactness of Hankel operators on weighted Fock spaces. As an application, for bounded symbols, we show that the Hankel operator is compact if and only if is compact, which complements the classical compactness result of Berger and Coburn. Motivated by recent work of Bauer, Coburn, and Hagger, we also apply our results to the Berezin-Toeplitz quantization.

Paper Structure

This paper contains 21 sections, 25 theorems, 245 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Approach
  4. Preliminaries
  5. Weighted Fock spaces
  6. The Bergman projection
  7. Hankel operators
  8. Lattices in $\mathbb{C}^n$
  9. Fock Carleson measures
  10. Differential forms and an auxiliary integral operator
  11. The space $\mathrm{IDA}$
  12. Definitions and preliminary lemmas
  13. The decomposition
  14. $\mathop{\mathrm{IDA}}\nolimits$ as a Banach space
  15. Proof of Theorem \ref{['main1']}
  16. ...and 6 more sections

Key Result

Theorem \oldthetheorem

Let $f\in \mathcal{S}$ and suppose that $\mathrm{Hess}_{\mathbb R}\varphi \simeq \mathrm{E}$ as in weights. (a) For $0< p\leq q<\infty$ and $q\ge 1$, $H_f : F^p(\varphi)\to L^q(\varphi)$ is bounded if and only if $f\in \mathop{\mathrm{BDA}}\nolimits^q$, and $H_f$ is compact if and only if $f\in VDA^ (b) For $1\le q<p <\infty$, $H_f: F^p(\varphi)\to L^q(\varphi)$ is bounded if and only if it is com

Theorems & Definitions (56)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Corollary \oldthetheorem
  • proof
  • ...and 46 more