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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.
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
Full Text: DOI
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