Xia's Theorem for the Fock space $H^{2}(\mathbb{C}^n, d\mu)$

TL;DR

This paper provides a detailed proof of Xia's theorem, establishing that the C*-algebra generated by weakly localized operators on the Fock space $H^{2}(C^n, d\mu)$ is exactly $\mathcal{T}^{(1)}$, clarifying the algebraic structure of these operators.

Contribution

The paper offers a rigorous proof of Xia's theorem, confirming the structure of the C*-algebra generated by weakly localized operators on the Fock space.

Findings

The C*-algebra generated by weakly localized operators equals $\\mathcal{T}^{(1)}$.

Provides a detailed proof of Xia's theorem.

Clarifies the algebraic structure of operators on Fock space.

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 $H^{2}(\mathbb{C}^n, d\mu)$ coincides with $\mathcal{T}^{(1)}$.