Compact composition operators on Bloch type spaces (Q2889736)

Let \(\mu\) be a bounded, continuous and strictly positive weight on the unit disc \(\mathbb{D}\) of the complex plane. The \(\mu\)-Bloch space \(\mathcal{B}^{\mu}\) consists of those analytic functions \(f\) on \(\mathbb{D}\) for which \(\sup_{z \in \mathbb{D}} \mu(z) |f'(z)| < \infty\). When \(\mu(z)=1-|z|^2\), then \(\mathcal{B}^{\mu} = \mathcal{B}\) is the usual Bloch space. In the case \(\mu(z)=(1-|z|^2)^{\alpha}\), \(\alpha >0\), the space \(\mathcal{B}^{\mu}\) is denoted by \(\mathcal{B}^{\alpha}\). Extending recent work by \textit{H. Wulan}, \textit{D. Zheng} and \textit{K. Zhu} [Proc. Am. Math. Soc. 137, No. 11, 3861--3868 (2009; Zbl 1194.47038)] and \textit{R. Zhao} [Proc. Am. Math. Soc. 138, No. 7, 2537--2546 (2010; Zbl 1190.47028)], the authors characterize continuity and compactness of composition operators from \(\mathcal{B}^{\alpha}\) into \(\mathcal{B}^{\mu}\). The proofs are standard. More general results with a different point of view have been obtained independently by \textit{O. Hyvärinen} et al. in [Integral Equations Oper. Theory 72, No. 2, 151--157 (2012; Zbl 1252.47026)].