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

Daniel Li (1)

1 LML - Laboratoire de Mathématiques de Lens

It is known, from results of B. MacCluer and J. Shapiro (1986), that every composition operator which is compact on the Hardy space $H^p$, $1 \leq p < \infty$, is also compact on the Bergman space ${\mathfrak B}^p = L^p_a (\D)$. In this survey, after having described the above known results, we consider Hardy-Orlicz $H^\Psi$ 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 $H^\Psi$ 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

Domains: Mathematics [math]; Functional Analysis [math.FA]

Daniel Li. Compact composition operators on Hardy-Orlicz and Bergman-Orlicz spaces. 2011. ⟨hal-00530387v2⟩