Compactness of operators on generalized Fock spaces

TL;DR

This paper characterizes when Toeplitz operators on generalized Fock spaces are compact, providing necessary and sufficient conditions, and shows these operators can be approximated by Toeplitz operators with smooth compactly supported symbols.

Contribution

It establishes a comprehensive criterion for compactness of Toeplitz operators on broad classes of weighted Fock spaces and demonstrates their approximation by smooth-symbol Toeplitz operators.

Findings

Necessary and sufficient conditions for compactness of Toeplitz operators.

All compact operators are in the norm closure of Toeplitz operators with smooth compactly supported symbols.

In the Hilbert space setting, compact operators can be approximated by Toeplitz operators with smooth symbols.

Abstract

For a very general class of weighted Fock spaces on $\mathbb{C}^n$, we give necessary and sufficient conditions for a Toeplitz operator with a (not necessarily positive) measure symbol to be compact. Furthermore, we show that all compact operators are in the norm closure of the algebra generated by Toeplitz operators with $C_c ^\infty(\mathbb{C}^n)$ symbols, and in the Hilbert space setting show that all compact operators are in the norm closure of the set of such Toeplitz operators.