Localization and compactness of Operators on Fock Spaces

Abstract

For $0<p\le \infty$, let $F_{\phi }^p$ be the Fock space induced by a weight function $\phi$ satisfying $d{d}^c\phi \simeq {\omega }_0$. In this paper, given $p\in (0,1]$ we introduce the concept of weakly localized operators on $F_{\phi }^p$, we characterize the compact operators in the algebra generated by weakly localized operators. As an application, for $0<p<\infty$ we prove that an operator $T$ in the algebra generated by bounded Toeplitz operators with $\text{BMO}$ symbols is compact on $F_{\phi }^p$ if and only if its Berezin transform satisfies certain vanishing property at $\infty$. In the classical Fock space, we extend the Axler-Zheng condition on linear operators $T$, which ensures $T$ is compact on $F_{\alpha }^p$ for all possible $0<p<\infty$.

23 Pages