Compact composition operators on Hardy-Orlicz and Bergman-Orlicz spaces

TL;DR

This paper explores the compactness of composition operators on Hardy-Orlicz and Bergman-Orlicz spaces, revealing new distinctions and characterizations that extend classical results from Hardy and Bergman spaces.

Contribution

It provides a characterization of compactness for composition operators on Hardy-Orlicz and Bergman-Orlicz spaces and identifies cases where compactness differs between these spaces.

Findings

Compactness on Hardy-Orlicz does not imply compactness on Bergman-Orlicz for certain Orlicz functions.

Classical results for Hardy and Bergman spaces are extended to Orlicz spaces.

Existence of Orlicz functions with differing compactness properties for composition operators.

Abstract

It is known, from results of B. MacCluer and J. Shapiro (1986), that every composition operator which is compact on the Hardy space $H^p$, $1 \leq p < \infty$, is also compact on the Bergman space ${\mathfrak B}^p = L^p_a (\D)$. In this survey, after having described the above known results, we consider Hardy-Orlicz $H^\Psi$ and Bergman-Orlicz ${\mathfrak B}^\Psi$ spaces, characterize the compactness of their composition operators, and show that there exist Orlicz functions for which there are composition operators which are compact on $H^\Psi$ but not on ${\mathfrak B}^\Psi$.