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arXiv · 2107.09819

Essential Commutants on Strongly Pseudo-convex Domains

Yi Wang·Jingbo Xia
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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.

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Yi Wang, Jingbo Xia. 2021-07-21. Essential Commutants on Strongly Pseudo-convex Domains. https://arxiv.org/abs/2107.09819

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