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 $L_a^2(\Omega )$. 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 $⟨A{k}_z,k_z⟩\to 0$ as $z\to \partial \Omega$, then $A$ is a compact operator.

60 pages; This version of the paper contains the technical details that were omitted in the published version, J. Funct. Anal. 280 (2021), no.1, 108775