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