Originally Published In
Journal of Mathematical Analysis and Applications Volume 461, Issue 2, 15 May 2018, Pages 1711-1732. https://doi.org/10.1016/j.jmaa.2017.12.046
For 0<p≤∞, let Fpφ be the Fock space induced by a weight function φ satisfying ddcφ≃ω0. In this paper, given p∈(0,1] we introduce the concept of weakly localized operators on Fpφ, 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 Fpφ 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 Fpα for all possible 0<p<∞.
HU, Zhangjian; Lv, Xiaofen; and Wick, Brett D., "Localization and compactness of operators on Fock spaces" (2018). Mathematics Faculty Publications. 46.