# Compact composition operators on Bergman-Orlicz spaces

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Abstract : We construct an analytic self-map $\phi$ of the unit disk and an Orlicz function $\Psi$ for which the composition operator of symbol $\phi$ is compact on the Hardy-Orlicz space $H^\Psi$, but not compact on the Bergman-Orlicz space ${\mathfrak B}^\Psi$. For that, we first prove a Carleson embedding theorem, and then characterize the compactness of composition operators on Bergman-Orlicz spaces, in terms of Carleson function (of order $2$). We show that this Carleson function is equivalent to the Nevanlinna counting function of order $2$.
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Cited literature [14 references]

https://hal-univ-artois.archives-ouvertes.fr/hal-00426831
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Submitted on : Tuesday, March 30, 2010 - 9:13:23 AM
Last modification on : Friday, January 21, 2022 - 3:30:44 AM
Long-term archiving on: : Thursday, September 23, 2010 - 12:11:28 PM

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### Identifiers

• HAL Id : hal-00426831, version 2
• ARXIV : 0910.5368

### Citation

Pascal Lefèvre, Daniel Li, Hervé Queffélec, Luis Rodriguez-Piazza. Compact composition operators on Bergman-Orlicz spaces. 2010. ⟨hal-00426831v2⟩

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