Localization and compactness of operators on Fock spaces

For 0<p≤∞ let F_φ ^p be the Fock space induced by a weight function φ satisfying dd^cφ≃ω₀. In this paper, given p∈(0,1] we introduce the concept of weakly localized operators on F_φ ^p, we characterize the compact operators in the algebra generated by weakly localized operators. As an application, for 0<p<∞ we prove that an operator T in the algebra generated by bounded Toeplitz operators with BMO symbols is compact on F_φ ^p if and only if its Berezin transform satisfies certain vanishing property at ∞. In the classical Fock space, we extend the Axler–Zheng condition on linear operators T, which ensures T is compact on F_α ^p for all possible 0<p<∞.