Multiplication operators on Orlicz and weighted Orlicz spaces

Abstract

Let $(\Omega ,\Sigma ,\mu )$ be a $\sigma$-finite complete measure space, $\tau :\Omega \to \Omega$ be a measurable transformation and $\varphi$ be an Orlicz function. In this article, first a necessary and sufficient condition for the bounded multiplication operator $M_u$ on Orlicz space $L^{\varphi }(\Omega )$ induced by measurable function $u$ to be completely continuous has been established. Next by using Radon-Nikodym derivative $\omega =\frac{d\mu \circ {\tau }^{-1}}{d\mu }$, the multiplication operator $M_u$ on weighted Orlicz space $L_{\omega }^{\varphi }(\Omega )$ have been characterized.