Some revisited results about composition operators on Hardy spaces

On the one hand, we generalize some results known for composition operators on Hardy spaces to the case of Hardy-Orlicz spaces H-Psi: construction of a "slow" Blaschke product giving a non-compact composition operator on H-Psi and yet "nowhere differentiable"; construction of a surjective symbol whose associated composition operator is compact on H-Psi and is, moreover, in all Schatten classes S-p(H-2), p > 0. On the other hand, we revisit the classical case of composition operators on H-2, giving first a new, and simpler, characterization of composition operators with closed range, and then showing directly the equivalence of the two characterizations of membership in Schatten classes of Luecking, and Luecking-Zhu.