Multiplication Operators on Generalized Orlicz Spaces Associated to Banach Function Spaces

In this paper, we study multiplication operators on generalized Orlicz spaces X Phi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X<^>\Phi$$\end{document} associated to a Banach function space X, where Phi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi$$\end{document} is a Young function, and give some characterization of them to be well-defined and bounded. Also, we present some sufficient and necessary conditions for such operators to be compact or invertible. Moreover, we find the essential norm of a multiplication operator on X Phi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X<^>\Phi$$\end{document} while the context measure space is discrete. Many results of this paper cover known Banach function spaces related to Orlicz one.