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Keywords:
Orlicz-Bochner space; property $(\beta )$; Orlicz space
Orlicz-Bochner space; property $(\beta )$; Orlicz space
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.
References:
[1] Alherk G., Hudzik H.: Uniformly non-$l_n^{(1)}$ Musielak-Orlicz spaces of Bochner type. Forum Math. 1 (1989), 403-410. MR 1016681
[2] Cerda J., Hudzik H., Mastyło M.: Geometric properties of Köthe Bochner spaces. Math. Proc. Cambridge Philos. Soc. 120 (1996), 521-533. MR 1388204
[3] Chen S., Hudzik H.: On some convexities of Orlicz and Orlicz-Bochner spaces. Comment. Math. Univ. Carolinae 29.1 (1988), 13-29. MR 0937545 | Zbl 0647.46030
[4] Clarkson J.A.: Uniformly convex spaces. Trans. Amer. Math. Soc. 40 (1936), 396-414. MR 1501880 | Zbl 0015.35604
[5] Cui Y., Płuciennik R., Wang T.: On property $(\beta)$ in Orlicz spaces. Arch. Math. 69 (1997), 57-69. MR 1452160 | Zbl 0894.46023
[6] Greim P.: Strongly exposed points in Bochner $L^p$ spaces. Proc. Amer. Math. Soc. 88 (1983), 81-84. MR 0691281
[7] Hudzik H.: Uniformly non-$l_n^{(1)}$ Orlicz spaces with Luxemburg norm. Studia Math. 81.3 (1985), 271-284. MR 0808569
[8] Hudzik H., Landes T.: Characteristic of convexity of Köthe function spaces. Math. Ann. 294 (1992), 117-124. MR 1180454 | Zbl 0761.46016
[9] Huff R.: Banach spaces which are nearly uniformly convex. Rocky Mountain J. Math. 10 (1980), 743-749. MR 0595102 | Zbl 0505.46011
[10] Kamińska A.: Uniform rotundity of Musielak-Orlicz sequence spaces. J. Approx. Theory 47 (1986), 302-322. MR 0862227
[11] Kamińska A.: Rotundity of Orlicz-Musielak sequence spaces. Bull. Acad. Polon. Sci. Math. 29 3-4 (1981), 137-144. MR 0638755
[12] Kolwicz P.: On property $(\beta)$ in Banach lattices, Calderón-Lozanowskiĭ and Orlicz-Lorentz spaces. submitted. Zbl 0993.46009
[13] Kolwicz P., Płuciennik R.: P-convexity of Bochner-Orlicz spaces. Proc. Amer. Math. Soc. 126.8 (1998), 2315-2322. MR 1443391
[14] Kutzarowa D.N.: An isomorphic characterization of property $(\beta)$ of Rolewicz. Note Mat. 10.2 (1990), 347-354. MR 1204212
[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
[16] Lin P.K.: Köthe Bochner Function Spaces. to appear. MR 2018062 | Zbl 1054.46003
[17] Montesinos V.: Drop property equals reflexivity. Studia Math. 87 (1987), 93-100. MR 0924764 | Zbl 0652.46009
[18] Płuciennik R.: On characterization of strongly extreme points in Köthe Bochner spaces. Rocky Mountain J. Math. 27.1 (1997), 307-315. MR 1453105
[19] Płuciennik R.: Points of local uniform rotundity in Köthe Bochner spaces. Arch. Math. 70 (1998), 479-485. MR 1621994
[20] Rolewicz S.: On drop property. Studia Math. 85 (1987), 27-35. MR 0879413
[21] Rolewicz S.: On $\Delta $-uniform convexity and drop property. Studia Math. 87 (1987), 181-191. MR 0928575 | Zbl 0652.46010
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