Normal structure of Lorentz-Orlicz spaces

• Autorzy: Pei-Kee Lin , Huiying Sun
• Języki publikacji: EN

Abstrakty

EN

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 = sup{u: ϕ is linear on (0,u)}, v₀=sup{v: w is constant on (0,v)}
(where sup∅ = 0). We prove the following theorem.
Theorem. Suppose that $Λ_{ϕ,w}(0,∞)$ (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, $∫_^{v₀} ϕ(u₀) · w < 2 and u₀ <∞).
(2) $Λ_{ϕ,w}$ has weakly normal structure if and only if $∫_0^{v₀} ϕ(u₀)· w < 2$.

Słowa kluczowe

EN

Lorentz-Orlicz space; normal sturcture; order continuous; Young function

Kategorie tematyczne

• 46B20: Geometry and structure of normed linear spaces
• 46B42: Banach lattices

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 1997
• Tom: 67
• Numer: 2
• Strony: 147-168

Daty

wydano

1997

otrzymano

1996-08-30

poprawiono

1997-04-07

Twórcy

autor

Pei-Kee Lin

• Department of Mathematics, University of Memphis, Memphis, Tennessee 38152, U.S.A.

autor

Huiying Sun

• Department of Mathematics, Harbin Institute of Technology, Harbin, China

Bibliografia

• [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.
• [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.
• [3] S. Chen, Geometry of Orlicz spaces, Dissertationes Math. 356 (1996).
• [4] J. Diestel, Sequences and Series in Banach Spaces, Springer, 1984.
• [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.
• [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.
• [7] A. Kamińska, Some remarks on Orlicz-Lorentz spaces, Math. Nachr. 147 (1990), 29-38.
• [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.
• [9] W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004-1006.
• [10] T. Landes, Permanence properties of normal structure, Pacific J. Math. 110 (1984), 125-143.
• [11] T. Landes, Normal structure and weakly normal structure of Orlicz sequence spaces, Trans. Amer. Math. Soc. 285 (1984), 523-534.
• [12] P.-K. Lin and H. Y. Sun, Some geometric properties of Lorentz-Orlicz spaces, Arch. Math. (Basel) 64 (1995), 500-511.
• [13] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II, Springer, 1979.
• [14] M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Marcel Dekker, 1991.