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For $1 < p < \infty$ let $\mathcal{T}_p ^\alpha$ be the norm closure of thealgebra generated by Toeplitz operators with bounded symbols acting on thestandard weighted Fock space $F_\alpha ^p$. In this paper, we will show that anoperator $A$ 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.

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