Linear operators on non-locally convex Orlicz spaces
Volume 79 / 2008
Abstract
We study linear operators from a non-locally convex Orlicz space $L^\Phi$ to a Banach space $(X,\|\cdot\|_X)$. Recall that a linear operator $T:L^\Phi\to X$ is said to be $\sigma$-smooth whenever $u_n\mathrel{\mathop{\longrightarrow}\limits^{{({\rm o})}}} 0$ in $L^\Phi$ implies $\|T(u_n)\|_X\to 0$. It is shown that every $\sigma$-smooth operator $T:L^\Phi\to X$ factors through the inclusion map $j:L^\Phi\to L^{{\overline{\Phi\hskip-.3pt}\hskip.3pt}}$, where ${\overline{\Phi\hskip-.3pt}\hskip.3pt}$ denotes the convex minorant of $\Phi$. We obtain the Bochner integral representation of $\sigma$-smooth operators $T:L^\Phi\to X$. This extends some earlier results of J. J. Uhl concerning the Bochner integral representation of linear operators defined on a locally convex Orlicz space.
