On some questions about composition operators on weighted Hardy spaces

Abstract

We first consider some questions raised by N. Zorboska in her thesis. In particular she asked for which sequences ${\hat{I}}^2$ every symbol $\ddot{I}†:\mathbb{D}\to \mathbb{D}$ with $\ddot{I}†\in H^2({\hat{I}}^2)$ induces a bounded composition operator $C_{\ddot{I}}•$ on the weighted Hardy space $H^2({\hat{I}}^2)$. We give partial answers and investigate when $H^2({\hat{I}}^2)$ is an algebra. We answer negatively another question in showing that there are a sequence ${\hat{I}}^2$ and $\ddot{I}†\in H^2({\hat{I}}^2)$ such that $\Vert \ddot{I}†{\Vert }_{\infty }<1$ and the composition operator $C_{\ddot{I}}†$ is not bounded on $H^2({\hat{I}}^2)$. In a second part, we show that for $p\ne 2$, no automorphism of $\mathbb{D}$, except those that fix $0$, induces a bounded composition operator on the Beurling-Sobolev space ${\ell }_A^p$, and even on any weighted version of this space.