Composition operator on Bers-type spaces.
Let \(D\) be the unit disk and let \(A(D)\) denote the set of analytic functions on \(D\). For \(\alpha\in (0,\infty)\), let \[ \begin{aligned} H^\infty_\alpha &= \Biggl\{f\in A(D)\,\biggl|\, \sup_{z\in D} (1-| z|)^\alpha| f(z)|< \infty\Biggr\},\\ H^\infty_{\alpha,0} &= \Biggl\{f\in A(D)\,\biggl|\, \lim_{| z|\to 1}(1- | z|)^\alpha| f(z)|= 0\Biggr\},\end{aligned} \] \(H^\infty_\alpha\) (or \(H^\infty_{\alpha,0}\)) is called Bers-type space. The goal of the paper under review is to give necessary and sufficient conditions for a composition operator to be bounded or compact on \(H^\infty_\alpha\) and \(H^\infty_{\alpha,0}\).
scientific article; zbMATH DE number 1824211