On some questions about composition operators on weighted Hardy spaces

We first consider some questions raised by N. Zorboska in her thesis. In particular she asked for which sequences $\beta$ every symbol $\phi \text{\hspace{0.17em}⁣}:\mathbb{D}\to \mathbb{D}$ with $\phi \in H^2(\beta )$ induces a bounded composition operator $C_{\phi }$ on the weighted Hardy space $H^2(\beta )$. We give partial answers and investigate when $H^2(\beta )$ is an algebra. We answer negatively another question in showing that there are a sequence $\beta$ and $\phi \in H^2(\beta )$ such that $\mathrm{\parallel }\phi {\mathrm{\parallel }}_{\mathrm{\infty }}<1$ and the composition operator $C_{\phi }$ is not bounded on $H^2(\beta )$. In a second part, we show that for $p\mathrm{\ne }2$, no automorphism of $\mathbb{D}$, except those that fix $0$, induces a bounded composition operator on the Beurling-Sobolev space ${\mathrm{\ell }}_A^p$, and even on any weighted version of this space.