Some revisited results about composition operators on Hardy spaces

TL;DR

This paper extends known results about composition operators on Hardy spaces to Hardy-Orlicz spaces, introduces new characterizations for operators on H^2, and clarifies Schatten class memberships with simplified proofs.

Contribution

It generalizes results to Hardy-Orlicz spaces, constructs specific operators with unique properties, and simplifies characterizations of composition operators on H^2.

Findings

Constructed a non-compact composition operator on H^Ψ using a slow Blaschke product.

Built a surjective symbol with a compact composition operator in all Schatten classes.

Provided a new, simpler characterization of closed range composition operators on H^2.

Abstract

We generalize, on one hand, 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 $H^\Psi$; construction of a surjective symbol whose composition operator is compact on $H^\Psi$ and, moreover, is in all the 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 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.