Compactness characterization of operators in the Toeplitz algebra of the Fock space $F_\alpha ^p$

TL;DR

This paper characterizes compact operators in the Toeplitz algebra on weighted Fock spaces by linking their compactness to membership in the algebra and the vanishing of their Berezin transform at infinity.

Contribution

It provides a precise criterion for compactness of operators in the Toeplitz algebra on Fock spaces, connecting algebraic properties with Berezin transform behavior.

Findings

Operators are compact iff they belong to the Toeplitz algebra and their Berezin transform vanishes at infinity.

Provides a characterization of compactness in terms of algebra membership and Berezin transform.

Establishes a necessary and sufficient condition for compactness in this setting.

Abstract

For $1 < p < \infty$ let $\mathcal{T}_p ^\alpha$ be the norm closure of the algebra generated by Toeplitz operators with bounded symbols acting on the standard weighted Fock space $F_\alpha ^p$. In this paper, we will show that an operator $A$ is compact on $F_\alpha ^p$ if and only if $A \in \mathcal{T}_p ^\alpha$ and the Berezin transform $B_\alpha (A)$ of $A$ vanishes at infinity.