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Essentially hyponormal weighted composition operators on the Hardy and weighted Bergman spaces. (English) Zbl 07104710

Summary: Let \(\varphi\) be an analytic self-map of the open unit disk \(\mathbb{D}\) and let \(\psi\) be an analytic function on \(\mathbb{D}\) such that the weighted composition operator \(C_{\psi,\varphi}\) defined by \(C_{\psi,\varphi}(f)=\psi f\circ\varphi\) is bounded on the Hardy and weighted Bergman spaces. We characterize those weighted composition operators \(C_{\psi,\varphi}\) on \(H^2\) and \(A_{\alpha}^2\) that are essentially hypo-normal, when \(\varphi\) is a linear-fractional non-automorphism.

MSC:

47B33 Linear composition operators
47B20 Subnormal operators, hyponormal operators, etc.
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