On the one hand, we generalize some results known for composition operators on Hardy spaces to the case of Hardy–Orlicz spaces : construction of a “slow” Blaschke product giving a non-compact composition operator on and yet “nowhere differentiable”; construction of a surjective symbol whose associated composition operator is compact on~ and is, moreover, in all Schatten classes , . On the other hand, we revisit the classical case of composition operators on , 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.
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Pascal Lefèvre, Daniel Li, Hervé Queffélec, Luis Rodríguez Piazza, Some revisited results about composition operators on Hardy spaces. Rev. Mat. Iberoam. 28 (2012), no. 1, pp. 57–76