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

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 < infinity, is also compact on the Bergman space B-p = L-a(p)(D). In this survey, after having described the above known results, we consider Hardy-Orlicz H-Psi and Bergman-Orlicz 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 B-Psi.