Multiplication operators on Orlicz and weighted Orlicz spaces

TL;DR

This paper characterizes bounded and completely continuous multiplication operators on Orlicz and weighted Orlicz spaces, providing necessary and sufficient conditions using measure-theoretic tools.

Contribution

It establishes a complete characterization of multiplication operators on Orlicz and weighted Orlicz spaces, including conditions for boundedness and complete continuity.

Findings

Necessary and sufficient condition for complete continuity of $M_u$ on Orlicz spaces.

Characterization of $M_u$ on weighted Orlicz spaces using Radon-Nikodym derivative.

Results extend understanding of operator behavior in generalized function spaces.

Abstract

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.