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 Hp, 1 ≤ p < ∞, is also compact on the Bergman space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathfrak B}^p = L^{p}_{a} ({\mathbb D})}$$\end{document}. In this survey, after having described the above known results, we consider Hardy-Orlicz HΨ and Bergman-Orlicz \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathfrak B}^\Psi}$$\end{document} 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathfrak B}^\Psi}$$\end{document}.