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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 \({{\mathfrak B}^\Psi}\).

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Authors and Affiliations

  1. Univ Lille-Nord-de-France, 59000, Lille, France

    Daniel Li

  2. UArtois, Laboratoire de Mathématiques de Lens EA 2462, Fédération CNRS Nord-Pas-de-Calais FR 2956, Faculté des Sciences Jean Perrin, Rue Jean Souvraz, S.P., 18, 62300, Lens, France

    Daniel Li

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  1. Daniel Li
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Correspondence to Daniel Li.

Additional information

These results come from joint works with P. Lefèvre, H. Queffélec and L. Rodríguez-Piazza ([4–6]). 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.

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Li, D. Compact composition operators on Hardy-Orlicz and Bergman-Orlicz spaces. RACSAM 105, 247–260 (2011). https://doi.org/10.1007/s13398-011-0027-5

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  • DOI: https://doi.org/10.1007/s13398-011-0027-5

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