Boundedness and compactness of Hausdorff operators on Fock spaces

TL;DR

This paper provides a comprehensive analysis of the boundedness and compactness of Hausdorff operators on Fock spaces, including conditions for transforming between different Fock spaces and extensions to Banach spaces of entire functions.

Contribution

It offers a complete characterization of bounded and compact Hausdorff operators on Fock spaces, including necessary and sufficient conditions and extensions to mixed norm and weighted spaces.

Findings

Characterization of bounded Hausdorff operators on Fock spaces.

Conditions for compactness of Hausdorff operators.

Extension of results to Banach spaces of entire functions.

Abstract

We obtain a complete characterization of the bounded Hausdorff operators acting on a Fock space $F^p_\alpha$ and taking its values into a larger one $F^q_\alpha,\ 0 < p \leq q \leq \infty,$ as well as some necessary or sufficient conditions for a Hausdorff operator to transform a Fock space into a smaller one. Some results are written in the context of mixed norm Fock spaces. Also the compactness of Hausdorff operators on a Fock space is characterized. The compactness result for Hausdorff operators on the Fock space $F^\infty_\alpha$ is extended to more general Banach spaces of entire functions with weighted sup norms defined in terms of a radial weight and conditions for the Hausdorff operators to become $p$-summing are also included.