| Title:
|
The property ($\beta $) of Orlicz-Bochner sequence spaces (English) |
| Author:
|
Kolwicz, Paweł |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
42 |
| Issue:
|
1 |
| Year:
|
2001 |
| Pages:
|
119-132 |
| . |
| Category:
|
math |
| . |
| Summary:
|
A characterization of property $(\beta )$ of an arbitrary Banach space is given. Next it is proved that the Orlicz-Bochner sequence space $l_\Phi (X)$ has the property $(\beta )$ if and only if both spaces $l_\Phi $ and $X$ have it also. In particular the Lebesgue-Bochner sequence space $l_p(X)$ has the property $(\beta )$ iff $X$ has the property $(\beta )$. As a corollary we also obtain a theorem proved directly in [5] which states that in Orlicz sequence spaces equipped with the Luxemburg norm the property $(\beta )$, nearly uniform convexity, the drop property and reflexivity are in pairs equivalent. (English) |
| Keyword:
|
Orlicz-Bochner space |
| Keyword:
|
property $(\beta )$ |
| Keyword:
|
Orlicz space |
| MSC:
|
46B20 |
| MSC:
|
46B45 |
| MSC:
|
46E30 |
| MSC:
|
46E40 |
| idZBL:
|
Zbl 1056.46020 |
| idMR:
|
MR1825377 |
| . |
| Date available:
|
2009-01-08T19:08:46Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119228 |
| . |
| Reference:
|
[1] Alherk G., Hudzik H.: Uniformly non-$l_n^{(1)}$ Musielak-Orlicz spaces of Bochner type.Forum Math. 1 (1989), 403-410. MR 1016681 |
| Reference:
|
[2] Cerda J., Hudzik H., Mastyło M.: Geometric properties of Köthe Bochner spaces.Math. Proc. Cambridge Philos. Soc. 120 (1996), 521-533. MR 1388204 |
| Reference:
|
[3] Chen S., Hudzik H.: On some convexities of Orlicz and Orlicz-Bochner spaces.Comment. Math. Univ. Carolinae 29.1 (1988), 13-29. Zbl 0647.46030, MR 0937545 |
| Reference:
|
[4] Clarkson J.A.: Uniformly convex spaces.Trans. Amer. Math. Soc. 40 (1936), 396-414. Zbl 0015.35604, MR 1501880 |
| Reference:
|
[5] Cui Y., Płuciennik R., Wang T.: On property $(\beta)$ in Orlicz spaces.Arch. Math. 69 (1997), 57-69. Zbl 0894.46023, MR 1452160 |
| Reference:
|
[6] Greim P.: Strongly exposed points in Bochner $L^p$ spaces.Proc. Amer. Math. Soc. 88 (1983), 81-84. MR 0691281 |
| Reference:
|
[7] Hudzik H.: Uniformly non-$l_n^{(1)}$ Orlicz spaces with Luxemburg norm.Studia Math. 81.3 (1985), 271-284. MR 0808569 |
| Reference:
|
[8] Hudzik H., Landes T.: Characteristic of convexity of Köthe function spaces.Math. Ann. 294 (1992), 117-124. Zbl 0761.46016, MR 1180454 |
| Reference:
|
[9] Huff R.: Banach spaces which are nearly uniformly convex.Rocky Mountain J. Math. 10 (1980), 743-749. Zbl 0505.46011, MR 0595102 |
| Reference:
|
[10] Kamińska A.: Uniform rotundity of Musielak-Orlicz sequence spaces.J. Approx. Theory 47 (1986), 302-322. MR 0862227 |
| Reference:
|
[11] Kamińska A.: Rotundity of Orlicz-Musielak sequence spaces.Bull. Acad. Polon. Sci. Math. 29 3-4 (1981), 137-144. MR 0638755 |
| Reference:
|
[12] Kolwicz P.: On property $(\beta)$ in Banach lattices, Calderón-Lozanowskiĭ and Orlicz-Lorentz spaces.submitted. Zbl 0993.46009 |
| Reference:
|
[13] Kolwicz P., Płuciennik R.: P-convexity of Bochner-Orlicz spaces.Proc. Amer. Math. Soc. 126.8 (1998), 2315-2322. MR 1443391 |
| Reference:
|
[14] Kutzarowa D.N.: An isomorphic characterization of property $(\beta)$ of Rolewicz.Note Mat. 10.2 (1990), 347-354. MR 1204212 |
| Reference:
|
[15] Kutzarowa D.N., Maluta E., Prus S.: Property $(\beta)$ implies normal structure of the dual space.Rend. Circ. Math. Palermo 41 (1992), 335-368. MR 1230583 |
| Reference:
|
[16] Lin P.K.: Köthe Bochner Function Spaces.to appear. Zbl 1054.46003, MR 2018062 |
| Reference:
|
[17] Montesinos V.: Drop property equals reflexivity.Studia Math. 87 (1987), 93-100. Zbl 0652.46009, MR 0924764 |
| Reference:
|
[18] Płuciennik R.: On characterization of strongly extreme points in Köthe Bochner spaces.Rocky Mountain J. Math. 27.1 (1997), 307-315. MR 1453105 |
| Reference:
|
[19] Płuciennik R.: Points of local uniform rotundity in Köthe Bochner spaces.Arch. Math. 70 (1998), 479-485. MR 1621994 |
| Reference:
|
[20] Rolewicz S.: On drop property.Studia Math. 85 (1987), 27-35. MR 0879413 |
| Reference:
|
[21] Rolewicz S.: On $\Delta $-uniform convexity and drop property.Studia Math. 87 (1987), 181-191. Zbl 0652.46010, MR 0928575 |
| . |
