Abstract

Given an Orlicz NN-function φ\varphi and a positive decreasing weight ww, we present criteria for the diameter two property and for the Radon-Nikodým property in the Orlicz-Lorentz function and the sequence spaces Λφ,w\Lambda_{\varphi,w} and λφ,w\lambda_{\varphi,w}. We show that in the spaces Λφ,w\Lambda_{\varphi,w} and λφ,w\lambda_{\varphi,w}, equipped with the Luxemburg norm, the diameter of any relatively weakly open subset of the unit ball in these spaces is two if and only if φ\varphi does not satisfy the appropriate Δ2\Delta_2-condition, while they have the Radon-Nikodým property if and only if φ\varphi satisfies the appropriate Δ2\Delta_2-condition.

Author information

Contact details are reproduced from the original publication and may be historical.

Anna Kamińska

Dept. of Mathematical Sciences, University of Memphis, Memphis, TN 38152-3240, U.S.A.

kaminska@memphis.edu

Hyung-Joon Tag

Dept. of Mathematical Sciences, University of Memphis, Memphis, TN 38152-3240, U.S.A.

htag@memphis.edu

Keywords

  • Diameter two property
  • Radon-Nikodým property
  • Orlicz-Lorentz space

Mathematics Subject Classification

46B20, 46E30, 47B38

References

  1. T. Abrahamsen, V. Lima, O. Nygaard: Remarks on diameter 2 properties, J. Convex Analysis 20 (2013) 439–452.https://doi.org/10.68381/jca20026Google Scholar
  2. M. D. Acosta, J. Becerra-Guerrero: Slices in the unit ball of the symmetric tensor product of C(K) and L1 (µ), Ark. Mat. 47 (2009) 1–12.https://doi.org/10.1007/s11512-007-0057-6Google Scholar
  3. M. D. Acosta, A. Kamińska: Weak neighborhoods and the Daugavet property of the interpolation spaces L1 + L∞ and L1 ∩ L∞, Indiana Univ. Math. J. 57(1) (2008) 77–96.https://doi.org/10.1512/iumj.2008.57.3171Google Scholar
  4. M. Acosta, A. Kamińska, M. Mastyło: The Daugavet property and weak neighborhoods in Banach lattices, J. Convex Analysis 19(3) (2012) 875–912.https://doi.org/10.68381/jca19046Google Scholar
  5. J. Becerra-Gurrero, G. López-Pérez: Relatively weakly open subsets of the unit ball in function spaces, J. Math. Anal. Appl. 315 (2006) 544–554.https://doi.org/10.1016/j.jmaa.2005.06.063Google Scholar
  6. C. Bennett, R. Sharpley: Interpolation of Operators, Academic Press, Boston (1988).Google Scholar
  7. R. D. Bourgin: Geometric Aspects of Convex Sets with the Radon-Nikodým Property, Lecture Notes in Mathematics 993, Springer, Berlin (1983).https://doi.org/10.1007/BFb0069321Google Scholar
  8. S. Chen: Geometry of Orlicz Spaces, Dissertationes Mathematicae 356, Polish Academy of Sciences, Warszawa (1996).Google Scholar
  9. J. Diestel, J. J. Uhl: Vector Measures, Mathematical Surveys 15, American Mathematical Society, Providence (1977).https://doi.org/10.1090/surv/015Google Scholar
  10. G. A. Edgar, R. F. Wheeler: Topological properties of Banach spaces, Pac. J. Math. 115(2) (1984) 317–350.https://doi.org/10.2140/pjm.1984.115.317Google Scholar
  11. A. Kamińska: Some remarks on Orlicz-Lorentz spaces, Math. Nachr. 147 (1990) 29–38.https://doi.org/10.1002/mana.19901470104Google Scholar
  12. A. Kamińska, K. Leśnik, Y. Raynaud: Dual spaces to Orlicz-Lorentz spaces, Stud. Math. 222(3) (2014) 229–261.https://doi.org/10.4064/sm222-3-3Google Scholar
  13. A. Kamińska, Y. Raynaud: Isomorphic lp subspaces in Orlicz-Lorentz sequence spaces, Proc. Amer. Math. Soc. 134(8) (2006) 2317–2327.https://doi.org/10.1090/S0002-9939-06-08266-9Google Scholar
  14. A. Kamińska, Y. Raynaud: New formulas for decreasing rearrangements and a class of Orlicz-Lorentz spaces, Rev. Mat. Complut. 27(2) (2014) 587–621.https://doi.org/10.1007/s13163-013-0119-1Google Scholar
  15. L. V. Kantorovich, G. P. Akilov: Functional Analysis, Pergamon Press, Oxford (1982).Google Scholar
  16. M. A. Krasnoselskii, Ya. B. Rutickii: Convex Functions and Orlicz Spaces, P. Noordhoff, Groningen (1961).Google Scholar
  17. S. G. Krein, Ju. I. Petunin, E. M. Semenov: Interpolation of Linear Operators, Translations of Math. Monog. 54, AMS, Providence (1982).Google Scholar
  18. P. K. Lin: Köthe-Bochner Function Spaces, Birkhäuser, Boston (2004).https://doi.org/10.1007/978-0-8176-8188-3Google Scholar
  19. Bor-Luh Lin, Pei-Kee Lin, S. L. Troyanski: Characterizations of denting points, Proc. Amer. Math. Soc. 102 (1988) 526–528.https://doi.org/10.1090/S0002-9939-1988-0928972-1Google Scholar
  20. J. Lindenstrauss, L. Tzafriri: Classical Banach Spaces II, Springer, Berlin (1979).https://doi.org/10.1007/978-3-662-35347-9Google Scholar
  21. O. Nygaard, D. Werner: Slices in the unit ball of a uniform algebra, Arch. Math. 76(6) (2001) 441–444.https://doi.org/10.1007/PL00000455Google Scholar
  22. A. C. Zaanen: Integration, North-Holland, Amsterdam (1967).Google Scholar

Suggested citation

A. Kamińska, H.-J. Tag. “Diameter of Weak Neighborhoods and the Radon-Nikodým Property in Orlicz-Lorentz Spaces.” Journal of Convex Analysis 24 (2017), No. 3, 969–985. https://doi.org/10.68381/jca24058

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2017. Rights now held by Banach Press.