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.
@article{119228,
author = {Pawe\l\ Kolwicz},
title = {The property ($\beta $) of Orlicz-Bochner sequence spaces},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {42},
year = {2001},
pages = {119-132},
zbl = {1056.46020},
mrnumber = {1825377},
language = {en},
url = {http://dml.mathdoc.fr/item/119228}
}
Kolwicz, Paweł. The property ($\beta $) of Orlicz-Bochner sequence spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001) pp. 119-132. http://gdmltest.u-ga.fr/item/119228/
Uniformly non-$l_n^{(1)}$ Musielak-Orlicz spaces of Bochner type, Forum Math. 1 (1989), 403-410. (1989) | MR 1016681
Geometric properties of Köthe Bochner spaces, Math. Proc. Cambridge Philos. Soc. 120 (1996), 521-533. (1996) | MR 1388204
On some convexities of Orlicz and Orlicz-Bochner spaces, Comment. Math. Univ. Carolinae 29.1 (1988), 13-29. (1988) | MR 0937545 | Zbl 0647.46030
Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936), 396-414. (1936) | MR 1501880 | Zbl 0015.35604
On property $(\beta)$ in Orlicz spaces, Arch. Math. 69 (1997), 57-69. (1997) | MR 1452160 | Zbl 0894.46023
Strongly exposed points in Bochner $L^p$ spaces, Proc. Amer. Math. Soc. 88 (1983), 81-84. (1983) | MR 0691281
Uniformly non-$l_n^{(1)}$ Orlicz spaces with Luxemburg norm, Studia Math. 81.3 (1985), 271-284. (1985) | MR 0808569
Characteristic of convexity of Köthe function spaces, Math. Ann. 294 (1992), 117-124. (1992) | MR 1180454 | Zbl 0761.46016
Banach spaces which are nearly uniformly convex, Rocky Mountain J. Math. 10 (1980), 743-749. (1980) | MR 0595102 | Zbl 0505.46011
Uniform rotundity of Musielak-Orlicz sequence spaces, J. Approx. Theory 47 (1986), 302-322. (1986) | MR 0862227
Rotundity of Orlicz-Musielak sequence spaces, Bull. Acad. Polon. Sci. Math. 29 3-4 (1981), 137-144. (1981) | MR 0638755
On property $(\beta)$ in Banach lattices, Calderón-Lozanowskiĭ and Orlicz-Lorentz spaces, submitted. | Zbl 0993.46009
P-convexity of Bochner-Orlicz spaces, Proc. Amer. Math. Soc. 126.8 (1998), 2315-2322. (1998) | MR 1443391
An isomorphic characterization of property $(\beta)$ of Rolewicz, Note Mat. 10.2 (1990), 347-354. (1990) | MR 1204212
Property $(\beta)$ implies normal structure of the dual space, Rend. Circ. Math. Palermo 41 (1992), 335-368. (1992) | MR 1230583
Köthe Bochner Function Spaces, to appear. | MR 2018062 | Zbl 1054.46003
Drop property equals reflexivity, Studia Math. 87 (1987), 93-100. (1987) | MR 0924764 | Zbl 0652.46009
On characterization of strongly extreme points in Köthe Bochner spaces, Rocky Mountain J. Math. 27.1 (1997), 307-315. (1997) | MR 1453105
Points of local uniform rotundity in Köthe Bochner spaces, Arch. Math. 70 (1998), 479-485. (1998) | MR 1621994
On drop property, Studia Math. 85 (1987), 27-35. (1987) | MR 0879413
On $\Delta $-uniform convexity and drop property, Studia Math. 87 (1987), 181-191. (1987) | MR 0928575 | Zbl 0652.46010