Compactness characterization of operators in the Toeplitz algebra of the   Fock space $F_\alpha ^p$
  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. 
Wolfram Bauer and Joshua Isralowitz
math.FA math.CV