| Title:
|
Extreme compact operators from Orlicz spaces to $C(\Omega)$ (English) |
| Author:
|
Chen, Shutao |
| Author:
|
Wisła, Marek |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
34 |
| Issue:
|
1 |
| Year:
|
1993 |
| Pages:
|
63-77 |
| . |
| Category:
|
math |
| . |
| 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 $T:E^{\varphi }(\mu )\rightarrow C(\Omega )$ 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. (English) |
| Keyword:
|
extreme points |
| Keyword:
|
vector valued continuous functions |
| Keyword:
|
compact linear operators |
| Keyword:
|
Orlicz spaces |
| MSC:
|
03D55 |
| MSC:
|
03E30 |
| MSC:
|
03E70 |
| MSC:
|
03H05 |
| MSC:
|
46B20 |
| MSC:
|
46E30 |
| idZBL:
|
Zbl 0801.46027 |
| idMR:
|
MR1240204 |
| . |
| Date available:
|
2009-01-08T18:01:08Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118556 |
| . |
| Reference:
|
[1] Aubin J.P., Cellina A.: Differential Inclusions.Springer Verlag, Berlin, 1984. Zbl 0538.34007, MR 0755330 |
| Reference:
|
[2] Blumenthal R.M., Lindenstrauss J., Phelps R.R.: Extreme operators into $C(K)$.Pacific J. Math. 15 (1965), 747-756. Zbl 0141.32101, MR 0209862 |
| Reference:
|
[3] Clausing A., Papadopoulou S.: Stable convex sets and extreme operators.Math. Ann. 231 (1978), 193-203. MR 0467249 |
| Reference:
|
[4] Dunford N., Schwartz J.T.: Linear Operators I, General Theory.Pure Appl. Math., vol. 7, Interscience, New York, 1958. Zbl 0084.10402, MR 0117523 |
| Reference:
|
[5] Grząślewicz R.: Extreme points in $C(K,L^{\varphi }(\mu))$.Proc. Amer. Math. Soc. 98 (1986), 611-614. Zbl 0606.46019, MR 0861761 |
| Reference:
|
[6] Krasnosel'skii M.A., Rutickii Y.B.: Convex Functions and Orlicz Spaces.Nordhoff, Groningen, 1961. |
| Reference:
|
[7] Lao B.Y., Zhu X.: Extreme points of Orlicz spaces (in Chinese).J. Zhongshan University, no. 2 (1983), 27-36. |
| Reference:
|
[8] Lindenstrauss J., Tzafriri L.: Classical Banach Spaces II.Springer Verlag, Berlin-Heidelberg- New York, 1977. Zbl 0403.46022, MR 0500056 |
| Reference:
|
[9] Luxemburg W.A.J.: Banach Function Spaces.Thesis, Delft, 1955. Zbl 0162.44701, MR 0072440 |
| Reference:
|
[10] Michael E.: Continuous selections I.Ann. of Math. (2) 63 (1956), 361-382. Zbl 0071.15902, MR 0077107 |
| Reference:
|
[11] Morris P.D., Phelps R.R.: Theorems of Krein-Milman type for certain convex sets of operators.Trans. Amer. Math. Soc. 150 (1970), 183-200. Zbl 0198.46601, MR 0262804 |
| Reference:
|
[12] Musielak J.: Orlicz spaces and modular spaces.Lecture Notes in Math. 1034, Springer Verlag, 1983. Zbl 0557.46020, MR 0724434 |
| Reference:
|
[13] Orlicz W.: Über eine gewisse Klasse von Räumen vom Typus B.Bull. Intern. Acad. Pol., série A, Kraków (1932), 207-220. Zbl 0006.31503 |
| Reference:
|
[14] Papadopoulou S.: On the geometry of stable compact convex sets.Math. Ann. 229 (1977), 193-200. Zbl 0339.46001, MR 0450938 |
| Reference:
|
[15] Wang Zhuogiang: Extreme points of Orlicz sequence spaces (in Chinese).J. Daqing Oil College, no. 1 (1983), 112-121. |
| Reference:
|
[16] Wisła M.: Extreme points and stable unit balls in Orlicz sequence spaces.Archiv der Math. 56 (1991), 482-490. MR 1100574 |
| Reference:
|
[17] Wisła M.: A full description of extreme points in $C(Ømega , L^{\varphi }(\mu))$.Proc. of Amer. Math. Soc. 113 (1991), 193-200. MR 1072351 |
| Reference:
|
[18] Wu Congxin, Wang Tingfu, Chen Shutao, Wang Youwen: Geometry of Orlicz Spaces (in Chinese).Harbin Institute of Technology, Harbin, 1986. |
| Reference:
|
[19] Wu Congxin, Zhao Shanzhong, Chen Junao: On calculation of rotundity of Orlicz spaces (in Chinese).J. Harbin Inst. of Technology, no. 2 (1978), 1-12. |
| . |
