Extreme compact operators from Orlicz spaces to $C(\Omega)$

Chen, Shutao ; Wisła, Marek

Commentationes Mathematicae Universitatis Carolinae, Tome 34 (1993) no. 1, pp. 63-77.

Résumé

Let $E^{\varphi }(\mu )$ be the subspace of finite elements of an Orlicz space endowed with the Luxemburg norm. The main theorem says that a compact linear operator $T:E^{\varphi }(\mu )\rightarrow C(\Omega )$ is extreme if and only if $T^{\ast }\omega \in \operatorname{Ext}\, B((E^{\varphi }(\mu ))^{\ast })$ on a dense subset of $\Omega $, where $\Omega $ is a compact Hausdorff topological space and $\langle T^{\ast } \omega ,x\rangle=(T x)(\omega )$. This is done via the description of the extreme points of the space of continuous functions $C(\Omega ,L^{\varphi }(\mu ))$, $L^{\varphi }(\mu )$ being an Orlicz space equipped with the Orlicz norm (conjugate to the Luxemburg one). There is also given a theorem on closedness of the set of extreme points of the unit ball with respect to the Orlicz norm.

Détail

Classification : 03D55, 03E30, 03E70, 03H05, 46B20, 46E30
Keywords: extreme points; vector valued continuous functions; compact linear operators; Orlicz spaces