Some examples of compact composition operators on H2

We construct, in an essentially explicit way, various composition operators on H2 and study their compactness or their membership in the Schatten classes. We construct: non-compact composition operators on H2 whose symbols have the same modulus on the boundary of D as symbols whose composition operators are in various Schatten classes Sp with p>2; compact composition operators which are in no Schatten class but whose symbols have the same modulus on the boundary of D as symbols whose associated composition operators are in Sp for every p>2.