Multiplication operators on Orlicz and weighted Orlicz spaces

Let $(\Omega,\Sigma,\mu)$ be a $\sigma$-finite complete measure space, $\tau:\Omega\rightarrow\Omega$ be a measurable transformation and $\phi$ be an Orlicz function. In this article, first a necessary and sufficient condition for the bounded multiplication operator $M_u$ on Orlicz space $L^\phi (\Omega)$ 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 $L^{\phi}_\omega(\Omega)$ have been characterized.