Some revisited results about composition operators on Hardy spaces

Abstract

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.