Localization and compactness of Operators on Fock Spaces

For $0<p\leq\infty$, let $F^{p}_\varphi$ be the Fock space induced by a weight function $\varphi$ satisfying $ dd^c \varphi \simeq \omega_0$. In this paper, given $p\in (0, 1]$ we introduce the concept of weakly localized operators on $ F^{p}_\varphi$, 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 $\textrm{BMO}$ symbols is compact on $F^p_\varphi$ 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 $F^p_{\alpha}$ for all possible $0<p<\infty$.