Composition and multiplication operators between Orlicz function spaces

Composition operators and multiplication operators between two Orlicz function spaces are investigated. First, necessary and sufficient conditions for their continuity are presented in several forms. It is shown that, in general, the Radon-Nikodým derivative d(μ∘τ−1)dμ(s)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac{d(\mu\circ\tau^{-1})}{d\mu}(s)$\end{document} need not belong to L∞(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{\infty}(\Omega)$\end{document} to guarantee the continuity of the composition operator cτx(t)=x(τ(t))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$c_{\tau}x(t)=x(\tau(t))$\end{document} from LΦ(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{\Phi}(\Omega)$\end{document} into LΨ(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{\Psi}(\Omega)$\end{document}. Next, the problem of compactness of these operators is considered. We apply a compactness criterion in Orlicz spaces which involves compactness with respect to the topology of local convergence in measure and equi-absolute continuity in norm of all the elements of the set under consideration. In connection with this, we state some sufficient conditions for equi-absolute continuity of the composition operator cτ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$c_{\tau}$\end{document} and the multiplication operator Mw\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$M_{w}$\end{document} from one Orlicz space into another. Also the problem of necessary conditions is discussed.