Some revisited results about composition operators on Hardy spaces

Abstract

We generalize, on one hand, some results known for composition operators on Hardy spaces to the case of Hardy-Orlicz spaces HΨ: construction of a “slow” Blaschke product giving a non-compact composition operator on HΨ; construction of a surjective symbol whose composition operator is compact on HΨ and, moreover, is in all the Schatten classes Sp(H2), p > 0. On the other hand, we revisit the classical case of composition operators on H2, giving first a new, and simplier, characterization of closed range composition operators, and then showing directly the equivalence of the two characterizations of membership in the Schatten classes of Luecking and Luecking and Zhu.