Compactness and essential norm properties of operators on generalized Fock spaces

TL;DR

This paper systematically studies the compactness and essential norm properties of operators on generalized weighted Fock spaces, providing necessary and sufficient conditions for compactness and estimates on essential norms.

Contribution

It offers new criteria for operator compactness and essential norm estimates on a broad class of weighted Fock spaces, including Toeplitz operators with bounded symbols.

Findings

Derived strong necessary and sufficient conditions for compactness.

Established estimates for the essential norm of operators.

Discussed open problems and potential extensions to weighted Bergman spaces.

Abstract

The purpose of this paper is to systematically study compactness and essential norm properties of operators on a very general class of weighted Fock spaces over $\C$. In particular, we obtain rather strong necessary and sufficient conditions for a wide class of operators (which includes operators in the Toeplitz algebra generated by bounded symbols) to be compact and we obtain related estimates on the essential norm of such operators. Finally, we discuss interesting open problems related to our results, and in particular discuss the possibility of extending our results to other generally weighted Bergman spaces on the unit ball of $\C$.