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

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.

arXiv commentv1: 36 pages, no figures v2: 36 pages, no figures, typos corrected, submitted v3: 41 pages, no figures, typos corrected, major revisions made (Proposition 3.6 of v2 generalised significantly and introduction/closing remarks changed significantly), to appear in the Journal of Functional Analysis