Normal structure of Lorentz-Orlicz spaces

Pei-Kee Lin; Huiying Sun

Pei-Kee Lin ; Huiying Sun

Annales Polonici Mathematici, Tome 66 (1997), p. 147-168 / Harvested from The Polish Digital Mathematics Library

Résumé

Let ϕ: ℝ → ℝ₊ ∪ 0 be an even convex continuous function with ϕ(0) = 0 and ϕ(u) > 0 for all u > 0 and let w be a weight function. u₀ and v₀ are defined by u₀ = supu: ϕ is linear on (0,u), v₀=supv: w is constant on (0,v) (where sup∅ = 0). We prove the following theorem. Theorem. Suppose that $Λ_{ϕ, w} (0, \infty)$ (respectively, $Λ_{ϕ, w} (0, 1)$ ) is an order continuous Lorentz-Orlicz space. (1) $Λ_{ϕ, w}$ has normal structure if and only if u₀ = 0 (respectively, $\int^{v ₀} ϕ (u ₀) \cdot w < 2 a n d u ₀ < \infty) .$ (2) $Λ_{ϕ, w}$ has weakly normal structure if and only if $\int_{0}^{v ₀} ϕ (u ₀) \cdot w < 2$ .

Publié le : 1997-01-01

Zbl 0901.46014

EUDML-ID : urn:eudml:doc:270441

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     author = {Pei-Kee Lin and Huiying Sun},
     title = {Normal structure of Lorentz-Orlicz spaces},
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     volume = {66},
     year = {1997},
     pages = {147-168},
     zbl = {0901.46014},
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Pei-Kee Lin; Huiying Sun. Normal structure of Lorentz-Orlicz spaces. Annales Polonici Mathematici, Tome 66 (1997) pp. 147-168. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv67z2p147bwm/

Bibliographie

[000] [1] N. L. Carothers, S. J. Dilworth, C. J. Lennard and D. A. Trautman, A fixed point property for the Lorentz space $L_{p, 1} (μ)$ , Indiana Univ. Math. J. 40 (1991), 345-352. | Zbl 0736.47029

[001] [2] N. L. Carothers, R. Haydon and P.-K. Lin, On the isometries of the Lorentz function spaces, Israel J. Math. 84 (1993), 265-287. | Zbl 0799.46024

[002] [3] S. Chen, Geometry of Orlicz spaces, Dissertationes Math. 356 (1996).

[003] [4] J. Diestel, Sequences and Series in Banach Spaces, Springer, 1984.

[004] [5] S. J. Dilworth and Y.-P. Hsu, The uniform Kadec-Klee property for the Lorentz space $L_{w, 1}$ , J. Austral. Math. Soc. Ser. A 60 (1996), 7-17. | Zbl 0852.46030

[005] [6] D. V. van Dulst and V. D. de Valk, (KK)-properties, normal structure and fixed points of nonexpansive mappings in Orlicz sequence spaces, Canad. J. Math. 38 (1986), 728-750. | Zbl 0615.46016

[006] [7] A. Kamińska, Some remarks on Orlicz-Lorentz spaces, Math. Nachr. 147 (1990), 29-38. | Zbl 0742.46013

[007] [8] A. Kamińska, P.-K. Lin and H. Y. Sun, Uniformly normal structure of Orlicz-Lorentz spaces, in: Interaction between Functional Analysis, Harmonic Analysis, and Probability, N. Kalton, E. Saab and S. Montgomery-Smith (eds.), Lecture Notes in Pure and Appl. Math. 175, Dekker, New York, 1996, 229-238. | Zbl 0856.46015

[008] [9] W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004-1006. | Zbl 0141.32402

[009] [10] T. Landes, Permanence properties of normal structure, Pacific J. Math. 110 (1984), 125-143. | Zbl 0534.46015

[010] [11] T. Landes, Normal structure and weakly normal structure of Orlicz sequence spaces, Trans. Amer. Math. Soc. 285 (1984), 523-534. | Zbl 0594.46010

[011] [12] P.-K. Lin and H. Y. Sun, Some geometric properties of Lorentz-Orlicz spaces, Arch. Math. (Basel) 64 (1995), 500-511. | Zbl 0823.46019

[012] [13] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II, Springer, 1979. | Zbl 0403.46022

[013] [14] M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Marcel Dekker, 1991.