On a criterion for continuity and compactness of composition operators acting on \(\alpha\)-Bloch spaces (Q1942071)

For \(\alpha>0\), the \(\alpha\)-Bloch space on the unit disc \(\mathbb{D}\), denoted by \(\mathcal{B}^{\alpha}\), is the collection of analytic functions \(f\) such that \[ \left\| f\right\|_\alpha=\sup_{z\in\mathbb{D}}\left(1-| z|^2\right)^{\alpha}\left| f'(z)\right|<\infty. \] For a self-map \(\phi\) of the disc, the composition operator \(C_\phi\) is defined by \(C_\phi f=f\circ \phi\). Let \(a\in\mathbb{D}\) and \(\sigma_{a}(z)=(1-| a|)\left(\left(1-\overline{a}z\right)^{-\alpha}-1\right)\). The authors prove that the composition operator \(C_\phi\) is bounded on \(\mathcal{B}^\alpha\) if and only if \[ \sup_{a\in\mathbb{D}} \left\| \sigma_a\circ\phi\right\|_{\alpha}<\infty. \] For compactness, the authors demonstrate that a vanishing version of the above condition is necessary and sufficient. These results are generalizations of previous work of \textit{M. Tjani} [Compact composition operators on some Moebius invariant Banach spaces. Ph.\,D.\ Thesis, Michigan State University (1996); Trans. Am. Math. Soc. 355, No.~11, 4683--4698 (2003; Zbl 1045.47020)].