Localizations and Essential Commutant of Toeplitz Algebra on Polydisk

TL;DR

This paper establishes that the norm closure of Toeplitz operators on the polydisk coincides with the Toeplitz algebra, and characterizes its essential commutant, revealing structural properties similar to those in the unit ball case.

Contribution

It proves the equality of the Toeplitz algebra with the norm closure of certain operator families on the polydisk and characterizes the essential commutant of this algebra.

Findings

The norm closure of Toeplitz operators on the polydisk equals the Toeplitz algebra.

The essential commutant of the Toeplitz algebra is characterized by functions of vanishing oscillation plus compact operators.

The Toeplitz algebra satisfies the double commutant relation in the Calkin algebra.

Abstract

Usually, the norm closure of a family of operators is not equal to the $C^*$-algebra generated by this family of operators. But, similar with the Bergman space $L^2_a(\textbf{B}, dv)$ of the unit ball in $\mathbb{C}^n$, we show that the norm closure of $\{T_f : f\in L^{\infty}(\mathbb{D}, dv)\}$ on Bergman space $L^2_a(\mathbb{D}, dv)$ of the ploydisk $\mathbb{D}$ in $\mathbb{C}^n$ actually coincides with the Toeplitz algebra $\mathcal{T}(\mathbb{D})$. A key ingredient in the proof is the class of operators $\mathcal{D}$ recently introduced by Yi Wang and Jingbo Xia. In fact, as a by-product, we simultaneously proved that $\mathcal{T}(\mathbb{D})$ also coincides with $\mathcal{D}$. Based on these results, we further proved that the essential commutant of Toeplitz algebra $\mathcal{T}(\mathbb{D})$ equals to $\{T_g: g\in VO_{bdd}\} + \mathcal{K}$ where $VO_{bdd}$ is the collection of functions of vanishing oscillation on polydisk $\mathbb{D}$ and $\mathcal{K}$ denotes the collection of compact operators on $L^2_a(\mathbb{D}, dv)$. On the other hand, we also prove that the essential commutant of $\{T_g: g\in VO_{bdd}\}$ is $\mathcal{T}(\mathbb{D})$, which implies that image of $\mathcal{T}(\mathbb{D})$ in the Calkin algebra satisfies the double commutant relation: $\pi(\mathcal{T}(\mathbb{D}))=\pi(\mathcal{T}(\mathbb{D}))''$.