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

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 ${\mathfrak B}^\Psi$.
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Cited literature [16 references]

https://hal-univ-artois.archives-ouvertes.fr/hal-00530387
Contributor : Daniel Li <>
Submitted on : Monday, March 21, 2011 - 10:43:40 AM
Last modification on : Thursday, September 27, 2018 - 2:58:02 PM
Long-term archiving on: : Wednesday, June 22, 2011 - 9:53:28 AM

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### Identifiers

• HAL Id : hal-00530387, version 2
• ARXIV : 1010.6207

### Citation

Daniel Li. Compact composition operators on Hardy-Orlicz and Bergman-Orlicz spaces. 2011. ⟨hal-00530387v2⟩

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