# Compact Hankel Operators with Bounded Symbols

## TL;DR

This paper investigates the conditions for compactness of Hankel operators with bounded symbols across Hardy, Bergman, and Fock spaces, providing new proofs and extending known results to broader contexts.

## Contribution

It offers a new proof using limit operator techniques for the compactness characterization of Hankel operators on Fock spaces and extends the theory to all Fock-Banach spaces.

## Key findings

- Hankel operator $H_f$ is compact iff $H_{ar f}$ is compact on Fock spaces.
- Compactness of Hankel operators is space-independent in Hardy and Bergman spaces.
- New proof clarifies the role of bounded analytic functions in compactness results.

## Abstract

We discuss the compactness of Hankel operators on Hardy, Bergman and Fock spaces with focus on the differences between the three cases, and complete the theory of compact Hankel operators with bounded symbols on the latter two spaces with standard weights. In particular, we give a new proof (using limit operator techniques) of the result that the Hankel operator $H_f$ is compact on Fock spaces if and only if $H_{\bar f}$ is compact. Our proof fully explains that this striking result is caused by the lack of nonconstant bounded analytic functions in the complex plane (unlike in the other two spaces) and extends the result from the Fock-Hilbert space to all Fock-Banach spaces. As in Hardy spaces, we also show that the compactness of Hankel operators is independent of the underlying space in the other two cases.