Essential Commutants on Strongly Pseudo-convex Domains

TL;DR

This paper characterizes the essential commutants of Toeplitz algebras on strongly pseudo-convex domains in complex analysis, extending previous results from the unit ball to more general domains using new techniques.

Contribution

It extends the understanding of Toeplitz algebra commutants from the unit ball to all strongly pseudo-convex domains with smooth boundaries, introducing new methods.

Findings

The Toeplitz algebra and a specific algebra of operators are mutual essential commutants.

Operators with boundary behavior tending to zero are compact.

New techniques are developed for general strongly pseudo-convex domains.

Abstract

Consider a bounded strongly pseudo-convex domain $\Omega $ with a smooth boundary in $\mathbb{C}^n$. Let $\mathcal{T}$ be the Toeplitz algebra on the Bergman space $L^2_a(\Omega )$. That is, $\mathcal{T}$ is the $C^\ast $-algebra generated by the Toeplitz operators $\{T_f : f \in L^\infty (\Omega )\}$. Extending previous work in the special case of the unit ball, we show that on any such $\Omega $, $\mathcal{T}$ and $\{T_f : f \in {\text{VO}}_{\text{bdd}}\} + \mathcal{K}$ are essential commutants of each other. On a general $\Omega $ considered in this paper, the proofs require many new ideas and techniques. These same techniques also enable us to show that for $A \in \mathcal{T}$, if $\langle Ak_z,k_z\rangle \rightarrow 0$ as $z \rightarrow \partial \Omega $, then $A$ is a compact operator.