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Essential commutants on strongly pseudo-convex domains. (English) Zbl 07270891
Summary: Consider a bounded strongly pseudo-convex domain $$\Omega$$ with smooth boundary in $$\mathbf{C}^n$$. 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 the work [27,28] 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, where $$\mathcal{K}$$ is the collection of compact operators on $$L_a^2({\Omega})$$. 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 $$\langle A k_z, k_z \rangle \to 0$$ as $$z \to \partial {\Omega}$$, then $$A$$ is a compact operator.
##### MSC:
 47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators 47L80 Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.) 46L05 General theory of $$C^*$$-algebras 32A36 Bergman spaces of functions in several complex variables
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