Localization and Compactness in Bergman and Fock spaces

TL;DR

This paper investigates conditions under which operators on Bergman and weighted Bargmann-Fock spaces are compact, focusing on Berezin transforms and kernel norms, extending previous results to broader settings.

Contribution

It extends the characterization of compact operators using Berezin transforms and growth conditions to the Bergman space for 1<p<∞ and generalizes results in weighted Bargmann-Fock spaces.

Findings

Vanishing Berezin transform plus growth conditions imply compactness.

Reproducing kernel thesis holds for operators with similar growth conditions.

Results extend Xia and Zheng's work to broader spaces and p-values.

Abstract

In this paper we study the compactness of operators on the Bergman space of the unit ball and on very generally weighted Bargmann-Fock spaces in terms of the behavior of their Berezin transforms and the norms of the operators acting on reproducing kernels. In particular, in the Bergman space setting we show how a vanishing Berezin transform combined with certain (integral) growth conditions on an operator $T$ are sufficient to imply that the operator is compact. In the weighted Bargmann-Fock space setting we show that the reproducing kernel thesis for compactness holds for operators satisfying similar growth conditions. The main results extend the results of Xia and Zheng to the case of the Bergman space when $1 < p < \infty$, and in the weighted Bargmann-Fock space setting, our results provide new, more general conditions that imply the work of Xia and Zheng via a more familiar approach that can also handle the $1 < p < \infty$ case.