Extreme compact operators from Orlicz spaces to C ( Ω )

Shutao Chen; Marek Wisła

Commentationes Mathematicae Universitatis Carolinae (1993)

  • Volume: 34, Issue: 1, page 63-77
  • ISSN: 0010-2628

Abstract

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Let E ϕ ( μ ) 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 ϕ ( μ ) → C ( Ω ) is extreme if and only if T * ω ∈ Ext B ( ( E ϕ ( μ ) ) * ) on a dense subset of Ω , where Ω is a compact Hausdorff topological space and 〈 T * ω , x 〉 = ( T x ) ( ω ) . This is done via the description of the extreme points of the space of continuous functions C ( Ω , L ϕ ( μ ) ) , L ϕ ( μ ) 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.

How to cite

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Chen, Shutao, and Wisła, Marek. "Extreme compact operators from Orlicz spaces to $C(\Omega )$." Commentationes Mathematicae Universitatis Carolinae 34.1 (1993): 63-77. <http://eudml.org/doc/247528>.

@article{Chen1993,
abstract = {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.},
author = {Chen, Shutao, Wisła, Marek},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {extreme points; vector valued continuous functions; compact linear operators; Orlicz spaces; vector valued continuous functions; compact linear operators; Orlicz space endowed with the Luxemburg norm; extreme points of the space of continuous functions; Orlicz space equipped with the Orlicz norm; closedness of the set of extreme points of the unit ball with respect to the Orlicz norm},
language = {eng},
number = {1},
pages = {63-77},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Extreme compact operators from Orlicz spaces to $C(\Omega )$},
url = {http://eudml.org/doc/247528},
volume = {34},
year = {1993},
}

TY - JOUR
AU - Chen, Shutao
AU - Wisła, Marek
TI - Extreme compact operators from Orlicz spaces to $C(\Omega )$
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1993
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 34
IS - 1
SP - 63
EP - 77
AB - 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.
LA - eng
KW - extreme points; vector valued continuous functions; compact linear operators; Orlicz spaces; vector valued continuous functions; compact linear operators; Orlicz space endowed with the Luxemburg norm; extreme points of the space of continuous functions; Orlicz space equipped with the Orlicz norm; closedness of the set of extreme points of the unit ball with respect to the Orlicz norm
UR - http://eudml.org/doc/247528
ER -

References

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