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

For $1<p<\mathrm{\infty }$ 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$andtheBerezintransform$B\alpha (A)$of$A$ vanishes at infinity.