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Localization operators on Bergman and Fock spaces

Pan Ma, Fugang Yan, Dechao Zheng, Kehe Zhu

Abstract

We introduce localization operators on weighted Bergman and Fock spaces and show that, under a natural scaling of symbols and window functions, localization operators on the weighted Bergman space $A_{βr^2}^2$ converge, in the weak sense, to localization operators on the Fock space $F_β^2$ as $r\to\infty$. From this we derive several applications, including one about sharp norm estimates for certain Toeplitz operators on Fock spaces, one about windowed Berezin transforms for weighted Bergman spaces, and another about Szegö-type theorems for localization operators on weighted Bergman spaces.

Localization operators on Bergman and Fock spaces

Abstract

We introduce localization operators on weighted Bergman and Fock spaces and show that, under a natural scaling of symbols and window functions, localization operators on the weighted Bergman space converge, in the weak sense, to localization operators on the Fock space as . From this we derive several applications, including one about sharp norm estimates for certain Toeplitz operators on Fock spaces, one about windowed Berezin transforms for weighted Bergman spaces, and another about Szegö-type theorems for localization operators on weighted Bergman spaces.
Paper Structure (5 sections, 28 theorems, 211 equations)

This paper contains 5 sections, 28 theorems, 211 equations.

Table of Contents

  1. Introduction
  2. The Fock space as a limit of weighted Bergman spaces
  3. Orthogonality relations in $A^2_{\alpha}$ and $F_{\beta}^2$
  4. Weak convergence of localization operators on $A^2_{\alpha}$
  5. Windowed Berezin transforms and applications

Key Result

Theorem 1.3

Suppose $\phi,\psi\in F_{\beta}^2$ with $\beta>0$ and $f\in L^{\infty}({\mathbb T}\times{\mathbb C})$. For any $\sigma\geq 0$ we have where $\phi_r(z)=\phi(rz)$ and

Theorems & Definitions (50)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Theorem 2.1
  • proof
  • ...and 40 more