Some examples of compact composition operators on $H^2$

Abstract

We construct, in an essentially explicit way, various composition operators on $H^2$ and study their compactness or their membership in the Schatten classes. We construct: non-compact composition operators on $H^2$ whose symbols have the same modulus on the boundary of $\mathbb{D}$ as symbols whose composition operators are in various Schatten classes $S_p$ with $p>2$; compact composition operators which are in no Schatten class but whose symbols have the same modulus on the boundary of $\mathbb{D}$ as symbols whose associated composition operators are in $S_p$ for every $p>2$.