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Weighted theory of Toeplitz operators on the Fock spaces

Jiale Chen

Abstract

We study the weighted compactness and boundedness of Toeplitz operators on the Fock spaces. Fix $α>0$. Let $T_{\varphi}$ be the Toeplitz operator on the Fock space $F^2_α$ over $\mathbb{C}^n$ with symbol $\varphi\in L^{\infty}$. For $1<p<\infty$ and any finite sum $T$ of finite products of Toeplitz operators $T_{\varphi}$'s, we show that $T$ is compact on the weighted Fock space $F^p_{α,w}$ if and only if its Berezin transform vanishes at infinity, where $w$ is a restricted $A_p$-weight on $\mathbb{C}^n$. Concerning boundedness, for $1\leq p<\infty$, we characterize the $r$-doubling weights $w$ such that $T_{\varphi}$ is bounded on the weighted spaces $L^p_{α,w}$ via a $\varphi$-adapted $A_p$-type condition. Our method also establishes a two weight inequality for the Fock projections in the case of $r$-doubling weights. Moreover, we characterize the corresponding weighted compactness of Bergman--Toeplitz operators, which answers a question raised by Stockdale and Wagner [Math. Z. 305 (2023), no. 1, Paper No. 10].

Weighted theory of Toeplitz operators on the Fock spaces

Abstract

We study the weighted compactness and boundedness of Toeplitz operators on the Fock spaces. Fix . Let be the Toeplitz operator on the Fock space over with symbol . For and any finite sum of finite products of Toeplitz operators 's, we show that is compact on the weighted Fock space if and only if its Berezin transform vanishes at infinity, where is a restricted -weight on . Concerning boundedness, for , we characterize the -doubling weights such that is bounded on the weighted spaces via a -adapted -type condition. Our method also establishes a two weight inequality for the Fock projections in the case of -doubling weights. Moreover, we characterize the corresponding weighted compactness of Bergman--Toeplitz operators, which answers a question raised by Stockdale and Wagner [Math. Z. 305 (2023), no. 1, Paper No. 10].
Paper Structure (5 sections, 21 theorems, 120 equations)

This paper contains 5 sections, 21 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weighted compactness
  4. Weighted boundedness
  5. Weighted compactness on Bergman spaces

Key Result

Theorem 1

Let $\alpha>0$ and $1<p<\infty$. Suppose that $T$ is an operator in the norm closure of the algebra generated by Toeplitz operators with $L^{\infty}$-symbols acting on $F^p_{\alpha}$. Then $T$ is compact on $F^p_{\alpha}$ if and only if $\lim_{|z|\to\infty}\widetilde{T}(z)=0$.

Theorems & Definitions (35)

  • Theorem 1: BI
  • Theorem 2: CFPIs14
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Example 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • ...and 25 more