On some questions about composition operators on weighted Hardy spaces (2311.01062v2)
Abstract: We first consider some questions raised by N. Zorboska in her thesis. In particular she asked for which sequences $\beta$ every symbol $\varphi \colon \mathbb{D} \to \mathbb{D}$ with $\varphi \in H2 (\beta)$ induces a bounded composition operator $C_\phi$ on the weighted Hardy space $H2 (\beta)$. We give partial answers and investigate when $H2 (\beta)$ is an algebra. We answer negatively another question in showing that there are a sequence $\beta$ and $\varphi \in H2 (\beta)$ such that $| \varphi |\infty < 1$ and the composition operator $C\varphi$ is not bounded on $H2 (\beta)$. In a second part, we show that for $p \neq 2$, no automorphism of $\mathbb{D}$, except those that fix $0$, induces a bounded composition operator on the Beurling-Sobolev space $\ellp_A$, and even on any weighted version of this space.