Operator norms and essential norms of weighted composition operators between Banach spaces of analytic functions
Let \(\mu\) be a weight, that is, a positive continuous function on the unit disk \(\mathbb{D}\). The authors characterize the bounded weighted composition operators \(W_{\psi, \varphi}\) from a general Banach space of analytic functions \(X\) into the growth space \(H^\infty_\mu\) and present an exact formula for the operator norm. Under certain assumptions on \(X\), they characterize the corresponding compact operators and estimate the essential norm of \(W_{\psi, \varphi}\) when \(X\) is a reflexive Banach space. Such results are applied to the Hardy spaces, the weighted Bergman spaces, and the Bloch space. Also, the authors obtain analogous results for the operators \(W_{\psi, \varphi}\) from \(X\) into the Bloch-type space \(B_\mu\). Reviewer's remark. A similar axiomatic approach was earlier used by the reviewer [J. Math. Sci., New York 182, No. 5, 630--638 (2012); translation from Zap. Nauchn. Semin. POMI 389, 85--100 (2011; Zbl 1257.30065)] to characterize the bounded operators \(W_{\psi, \varphi}\) from \(X\) into the Lipschitz-type spaces of arbitrary smoothness.
scientific article; zbMATH DE number 6506791
scientific article