Compact composition operators on general weighted spaces (Q2747332)

The author studies composition operators on weighted spaces \(H^\infty_v\) consisting of all analytic functions on the open unit disc \(f:\mathbb D\to C\) such that \(\sup_{z\in\mathbb D}v(z)|f(z)|<\infty\), where \(v\) is positive and continuous. The main result is a characterization of those weights \(v\in C(\overline{\mathbb D})\) with the property that there exists a compact operator \(C_\varphi\) (i.e. the composition operator with symbol \(\varphi\)) on \(H^\infty_v\) such that \(\overline{\varphi({\mathbb D})}\cap\partial{\mathbb D}\not=\emptyset\). The necessary and sufficient condition is that \(v\) has to vanish at one boundary point of \(\mathbb D\). The author does not assume the weights to be radial.