Xia's Theorem for the Fock space $H^{2}(\mathbb{C}^n, dμ)$
Solange Bridgitte Difo
TL;DR
This paper proves Xia's theorem for the Fock space $H^{2}(\C^{n},d\mu)$ by developing a localization framework for operators: defining weakly localized and XZ-sufficiently localized operators, and the Toeplitz-operator-based algebra $\mathcal{T}^{(1)}$. The core approach expresses any weakly localized operator as a norm-limit of localized pieces using the kernels $E_w$ and $E_z$, admissible decompositions, and norm-continuity arguments, culminating in $C^{*}(\mathcal{WL})=\mathcal{T}^{(1)}$. The paper also derives a compactness criterion: an operator is compact on $H^{2}(\C^{n},d\mu)$ iff it belongs to $C^{*}(\mathcal{WL})$ and its Berezin transform vanishes at infinity, tying localization to Berezin asymptotics. These results extend Xia's localization program from Bergman spaces to the Fock space, yielding a precise structural description of the operator algebra generated by weakly localized operators and practical compactness tests. The findings have implications for Toeplitz operator theory and the study of localized operator algebras in reproducing kernel Hilbert spaces.
Abstract
In this paper, we provide a detailed proof for Xia's following theorem: the C^{*}-algebra generated by the class of weakly localized operators on $H^{2}(\mathbb{C}^n, dμ)$ coincides with $\mathcal{T}^{(1)}$.