Localization and compactness of Operators on Fock Spaces
Zhangjian Hu, Xiaofen Lv, Brett D. Wick

TL;DR
This paper characterizes compact operators on Fock spaces induced by certain weights, introduces weakly localized operators, and extends classical compactness criteria to a broader class of operators and spaces.
Contribution
It introduces the concept of weakly localized operators on Fock spaces for p in (0,1], and characterizes their compactness within the generated algebra, extending classical results.
Findings
Characterization of compact operators via Berezin transform vanishing.
Introduction of weakly localized operators for p in (0,1].
Extension of Axler-Zheng condition to all p in (0,∞).
Abstract
For , let be the Fock space induced by a weight function satisfying . In this paper, given we introduce the concept of weakly localized operators on , we characterize the compact operators in the algebra generated by weakly localized operators. As an application, for we prove that an operator in the algebra generated by bounded Toeplitz operators with symbols is compact on 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 , which ensures is compact on for all possible .
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Taxonomy
TopicsHolomorphic and Operator Theory · Advanced Topics in Algebra · Advanced Operator Algebra Research
