Eigenvalues of weighted composition operators on the Bloch space (Q989592)

Let \(\mathbb{D}\) denote the open unit disk in the complex plane and \(H(\mathbb{D})\) the collection of all analytic functions on \(\mathbb{D}\). Through composition an analytic self-map \(\varphi\) of \(\mathbb{D}\) induces the classical linear composition operator \(C_{\phi}\) defined by \[ C_{\phi}: H(\mathbb{D}) \to H(\mathbb{D}), \; f \mapsto f \circ \phi. \] These objects are interesting form different points of view. For example, composition operators link complex analysis and operator theory, see, e.g., the excellent monographs of [\textit{C. C. Cowen} and \textit{B. D. MacCluer}, Composition operators on spaces of analytic functions. Studies in Advanced Mathematics. Boca Raton, FL: CRC Press (1995; Zbl 0873.47017)] and of [\textit{J. H. Shapiro}, Composition operators and classical function theory. Universitext: Tracts in Mathematics. New York: Springer-Verlag (1993; Zbl 0791.30033)]. Combining composition operators and pointwise multiplication by a map \(u \in H(\mathbb{D})\) we obtain the weighted composition operators \[ uC_{\phi}: H(\mathbb{D}) \to H(\mathbb{D}), \; f \mapsto u(f \circ \phi). \] Motivated by the work of [\textit{C. Hammond}, On the norm of a composition operator, Ph.\,D.\ dissertation, Graduate Faculty of the University of Virginia (2003)], under some assumptions on the map \(u\) the authors determine the set of eigenvalues of the operator \(u C_{\phi}\) acting on the Bloch space \[ {\mathcal B}:= \{ f \in \mathbb{D}: \sup_{z \in \mathbb{D}} (1-|z|^2) |f'(z)| < \infty \}. \]