On Bloch-to-Besov composition operators
Let \(H(D)\) denote the space of analytic functions defined on the open unit disk \(D\). Every analytic self-map \(\phi: D \to D\) induces a linear operator \(C_{\phi}: H(D) \to H(D)\), defined by \(C_{\phi}f=f\circ \phi\), \(f \in H(D)\). Let \(\mathcal B\), \(B_p\) and \(B_p^h\), \(1<p<\infty\), denote the Bloch space, the Besov space and the hyperbolic Besov space, respectively. The two theorems proved in this paper are as follows. Theorem 1. Let \(\phi: D \to D\) be an analytic self-map and let \(1<p<\infty\). Then \(C_{\phi}\) is a Bloch-to-\(B_p\) composition operator if and only if \(\phi \in B_p^h\). Theorem 2. If \(\phi \in B_p^h\), \(1<p<\infty\), then \(\phi\) induces a compact composition operator \(C_{\phi}\) on \(\mathcal B\) into \(B_p\).
scientific article; zbMATH DE number 1022277