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Journal of Operator Theory

Volume 69, Issue 2, Spring 2013  pp. 463-481.

Compact composition operators on the Hardy-Orlicz and weighted Bergman-Orlicz spaces on the ball

Authors Stephane Charpentier
Author institution: Departement de Mathematiques, Batiment 425, Universite Paris-Sud, F-91405, Orsay, France

Summary:  Using recent characterizations of the compactness of composition operators on the Hardy--Orlicz and Bergman--Orlicz spaces on the ball \cite{Charp1}, \cite{Charp2}, we first show that a composition operator which is compact on every Hardy--Orlicz (or Bergman--Orlicz) space has to be compact on $H^{\infty}$. Then, although it is well-known that a map whose range is contained in some nice Kor\'anyi approach region induces a compact composition operator on $H^{p} (\mathbb{B}_{N} )$ or on $A_{\alpha}^{p} (\mathbb{B}_{N} )$, we prove that, for each Kor\'anyi region $\Gamma$, there exists a map $\phi:\mathbb{B}_{N} \to \Gamma$ such that $C_{\phi}$ is not compact on $H^{\psi} (\mathbb{B}_{N} )$, when $\psi$ grows fast. Finally, we extend (and simplify the proof of) a result by K. Zhu for the classical weighted Bergman spaces, by showing that, under reasonable conditions, a composition operator $C_{\phi}$ is compact on the weighted Bergman--Orlicz space $A_{\alpha}^{\psi} (\mathbb{B}_{N} )$, if and only if\[ \lim_{ |z | \to 1}\frac{\psi^{-1} (1/ (1- |\phi (z ) | )^{N (\alpha )} )}{\psi^{-1} (1/ (1- |z | )^{N (\alpha )} )}=0.\] In particular, we deduce that the compactness of composition operators on $A_{\alpha}^{\psi} (\mathbb{B}_{N} )$ does not depend on $\alpha$ anymore when the Orlicz function $\psi$ grows fast.

Keywords:  Carleson measure, composition operator, Hardy--Orlicz space, several complex variables, weighted Bergman-Orlicz space

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