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Compact composition operators on Bergman-Orlicz spaces


Authors: Pascal Lefèvre, Daniel Li, Hervé Queffélec and Luis Rodríguez-Piazza
Journal: Trans. Amer. Math. Soc. 365 (2013), 3943-3970
MSC (2010): Primary 47B33; Secondary 30J10, 30H10, 30J99, 46E15
DOI: https://doi.org/10.1090/S0002-9947-2013-05922-3
Published electronically: April 24, 2013
MathSciNet review: 3055685
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Abstract | References | Similar Articles | Additional Information

Abstract: We construct an analytic self-map $ \varphi $ of the unit disk and an Orlicz function $ \Psi $ for which the composition operator of symbol $ \varphi $ is compact on the Hardy-Orlicz space $ H^\Psi $, but not on the Bergman-Orlicz space $ {\mathfrak{B}}^\Psi $. For that, we first prove a Carleson embedding theorem and then characterize the compactness of composition operators on Bergman-Orlicz spaces, in terms of Carleson function (of order $ 2$). We show that this Carleson function is equivalent to the Nevanlinna counting function of order $ 2$.


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Additional Information

Pascal Lefèvre
Affiliation: Laboratoire de Mathématiques de Lens EA 2462, Fédération CNRS Nord-Pas-de-Calais FR 2956, Université Lille-Nord-de-France UArtois, F-62 300 Lens, France
Email: pascal.lefevre@euler.univ-artois.fr

Daniel Li
Affiliation: Laboratoire de Mathématiques de Lens EA 2462, Fédération CNRS Nord-Pas-de-Calais FR 2956, Université Lille-Nord-de-France UArtois, Faculté des Sciences Jean Perrin, Rue Jean Souvraz, S.P.\kern1mm 18, F-62 300 Lens, France
Email: daniel.li@euler.univ-artois.fr

Hervé Queffélec
Affiliation: Laboratoire Paul Painlevé U.M.R. CNRS 8524, Université Lille-Nord-de-France USTL, F-59655 Villeneuve D’Ascq Cedex, France
Email: Herve.Queffelec@univ-lille1.fr

Luis Rodríguez-Piazza
Affiliation: Facultad de Matemáticas, Departamento de Análisis Matemático & IMUS, Universidad de Sevilla, Apartado de Correos 1160, 41 080 Sevilla, Spain
Email: piazza@us.es

DOI: https://doi.org/10.1090/S0002-9947-2013-05922-3
Keywords: Bergman-Orlicz space, Carleson function, compactness, composition operator, Hardy-Orlicz space, Nevanlinna counting function
Received by editor(s): May 4, 2010
Published electronically: April 24, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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