Localization operators on Bergman and Fock spaces

TL;DR

This paper studies localization operators on Bergman and Fock spaces, showing their convergence under scaling and deriving applications like norm estimates and Szeg"o-type theorems.

Contribution

It introduces a framework for localization operators on weighted Bergman and Fock spaces and demonstrates their convergence, leading to new results in operator norm estimates and transform analysis.

Findings

Localization operators on Bergman spaces converge to those on Fock spaces as scale increases.

Derived sharp norm estimates for Toeplitz operators on Fock spaces.

Established Szeg"o-type theorems for localization operators on weighted Bergman 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 $A_{\beta r^2}^2$ converge, in the weak sense, to localization operators on the Fock space $F_{\beta}^2$ 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 Szeg\"{o}-type theorems for localization operators on weighted Bergman spaces.