Essential Commutants on 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.