Mirzakarimi, G.; Seddighi, K. Weighted composition operators on Bergman and Dirichlet spaces. (English) Zbl 0891.47018 Georgian Math. J. 4, No. 4, 373-383 (1997). 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. Cited in 14 Documents 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 Keywords:Carleson measure; angular derivative; Hilbert space of analytic functions; weighted composition operator; compactness; weighted Dirichlet spaces PDFBibTeX XMLCite \textit{G. Mirzakarimi} and \textit{K. Seddighi}, Georgian Math. J. 4, No. 4, 373--383 (1997; Zbl 0891.47018) Full Text: EuDML EMIS