Date: 01 Apr 2011

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

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Abstract

It is known, from results of MacCluer and Shapiro (Canad. J. Math. 38(4):878–906, 1986), that every composition operator which is compact on the Hardy space H p , 1 ≤ p < ∞, is also compact on the Bergman space \({{\mathfrak B}^p = L^{p}_{a} ({\mathbb D})}\) . In this survey, after having described the above known results, we consider Hardy-Orlicz H Ψ 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 Ψ but not on \({{\mathfrak B}^\Psi}\) .

These results come from joint works with P. Lefèvre, H. Queffélec and L. Rodríguez-Piazza ([46]). It is an expanded version of the conference I gave at the ICM satellite conference Functional Analysis and Operator Theory, held in Bangalore, India, 8–11 august 2010.