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Extreme compact operators from Orlicz spaces to \(C(\Omega)\). (English) Zbl 0801.46027

Summary: 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 \(T:E^ \varphi (\mu) \to C (\Omega)\) is extreme if and only if \(T^*\omega \in \text{Ext} B((E^ \varphi (\mu))^*)\) on a dense subset of \(\Omega\), where \(\Omega\) is a compact Hausdorff topological space and \(\langle T^* \omega,x \rangle = (Tx) (\omega)\). 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.

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46B20 Geometry and structure of normed linear spaces
47B07 Linear operators defined by compactness properties
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