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 Hp, 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 HΨ 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 HΨ but not on \({{\mathfrak B}^\Psi}\).

Keywords

Bergman spaces Bergman-Orlicz spaces Blaschke product Carleson function Carleson measure Compactness Composition operator Hardy spaces Hardy-Orlicz spaces Nevanlinna counting function 

Mathematics Subject Classification (2000)

Primary 47B33 Secondary 30H10 30H20 30J10 46E15 

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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Univ Lille-Nord-de-FranceLilleFrance
  2. 2.UArtois, Laboratoire de Mathématiques de Lens EA 2462, Fédération CNRS Nord-Pas-de-Calais FR 2956, Faculté des Sciences Jean PerrinLensFrance

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