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

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1 LML - Laboratoire de Mathématiques de Lens

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 ${\mathfrak B}^\Psi$.

Keywords : Bergman spaces Bergman-Orlicz spaces Blaschke product Carleson function Carleson measure compactness composition operator Hardy spaces Hardy-Orlicz spaces Nevanlinna counting function

Autor: Daniel Li -

Fuente: https://hal.archives-ouvertes.fr/

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