Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas

, 105:247

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

Survey

DOI: 10.1007/s13398-011-0027-5

Cite this article as:
Li, D. RACSAM (2011) 105: 247. doi:10.1007/s13398-011-0027-5

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 spacesBergman-Orlicz spacesBlaschke productCarleson functionCarleson measureCompactnessComposition operatorHardy spacesHardy-Orlicz spacesNevanlinna counting function

Mathematics Subject Classification (2000)

Primary 47B33Secondary 30H1030H2030J1046E15

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