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Weighted composition operators on Bergman and Dirichlet spaces. (English) Zbl 0891.47018

Summary: Let \(H(\Omega)\) denote a functional Hilbert space of analytic functions on a domain \(\Omega\). Let \(w: \Omega\to \mathbb{C}\) and \(\phi:\Omega\to \Omega\) be such that \(wf\circ\phi\) is in \(H(\Omega)\) for every \(f\) in \(H(\Omega)\). The operator \(wC_\phi\) given by \(f\to wf\circ\phi\) is called a weighted composition operator on \(H(\Omega)\). In this paper, we characterize such operators and those for which \((wC_\phi)^*\) is a composition operator. Compact weighted composition operators on some functional Hilbert spaces are also characterized. We give sufficient conditions for the compactness of such operators on weighted Dirichlet spaces.

MSC:

47B38 Linear operators on function spaces (general)
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
46E20 Hilbert spaces of continuous, differentiable or analytic functions
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