IDA and Hankel operators on Fock spaces

TL;DR

This paper introduces a new function space to analyze Hankel operators on weighted Fock spaces, providing a complete characterization of their boundedness and compactness, and exploring implications for Berezin-Toeplitz quantization.

Contribution

It defines the space IDA and uses it to characterize Hankel operators' boundedness and compactness on weighted Fock spaces, extending classical results.

Findings

Hankel operator compactness characterized by symbol properties

Hankel operators with bounded symbols are compact iff their conjugates are compact

Applications to Berezin-Toeplitz quantization

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.