Abstract

It is known [see R. M. Blumenthal, J. Lindenstrauss, R. R. Phelps, Extreme operators into C(K), Pacific Journal of Mathematics 15(3) (1965), 747-756] that a compact linear operator from a Banach space XX into the space of continuous functions C(Z,R)C(Z,\R) is extreme provided it is nice, i.e. T∗(Z)⊂Ext⁡B(X∗)T^{*}(Z)\subset \Ext B(X^{*}), where ZZ is a compact Hausdorff space and T∗:Z→X∗T^{*}: Z\to X^{*} is a continuous function defined by T∗(z)(x)=T(x)(z)T^{*}(z)(x)=T(x)(z). The nice operator condition can be weakened as long as the set of extreme points Ext⁡B(X∗)\Ext B(X^{*}) is closed, namely it suffices to assume than T∗(Z0)⊂Ext⁡B(X∗)T^{*}(Z_0)\subset \Ext B(X^{*}) for some dense subset Z0⊂ZZ_0\subset Z in that case. The aim of this paper is to characterize the closedness of the set of extreme points of the unit ball of Calderon-Lozanovskii spaces EφE_{\varphi} generated by the Köthe space EE and the Orlicz function φ\varphi. The main theorem of the paper (Theorem 2.12) gives conditions under which the closedness of the set Ext⁡B(Eφ)\Ext B(E_{\varphi}) is equivalent to the closedness of the set of extreme points of the unit ball of the corresponding Köthe space EE.

Author information

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

Ewa Kasior

Institute of Mathematics, University of Szczecin, Wielkopolska 15, 70–451 Szczecin 3, Poland

ekasior@univ.szczecin.pl

Marek Wisła

Faculty of Mathematics and Computer Science, Adam Mickiewicz University, ul. Umultowska 87, 61-614 Poznań, Poland

marek.wisla@amu.edu.pl

Keywords

  • Calderón-Lozanovskii spaces
  • extreme points
  • compact operators
  • Orlicz spaces
  • Köthe spaces

Mathematics Subject Classification

46B20, 46E30

References

  1. R. M. Blumenthal, J. Lindenstrauss, R. R. Phelps: Extreme operators into C(K), Pac. J. Math. 15(3) (1965) 747–756.https://doi.org/10.2140/pjm.1965.15.747Google Scholar
  2. J. Cerda, H. Hudzik, M. Mastyło: On the geometry of some Calderón-Lozanovskĭi spaces, Indag. Math., New Ser. 6(1) (1995) 35–49.https://doi.org/10.1016/0019-3577(95)98199-LGoogle Scholar
  3. S. Chen: Geometry of Orlicz spaces, Diss. Math. 356 (1996) 204 pp.Google Scholar
  4. Y. Cui, H. Hudzik, L. Duan, M. Wisła: Basic theory of p-Amemiya norm in Orlicz spaces: extreme points and rotundity in Orlicz spaces endowed with these norms, Nonlinear Anal. 69 (2008) 1796–1816.https://doi.org/10.1016/j.na.2007.07.026Google Scholar
  5. Y. Cui, H. Hudzik, J. Li, M. Wisła: Strongly extreme points in Orlicz spaces equipped with the p-Amemiya norm, Nonlinear Anal. 71 (2009) 6343–6364.https://doi.org/10.1016/j.na.2009.06.085Google Scholar
  6. T. Dominguez, H. Hudzik, G. Lopez, M. Mastyło, B. Sims: Complete characterizations of Kadec-Klee properties in Orlicz spaces, Houston J. Math. 29(4) (2003) 1027–1044.Google Scholar
  7. N. Dunford, J. T. Schwartz: Linear Operators. I: General Theory, Pure Appl. Math. 7, Interscience, New York (1958).Google Scholar
  8. P. Foralewski: On some geometric properties of generalized Calderón-Lozanovskĭi spaces, Acta Math. Hung. 80(1–2) (1998) 55–66.https://doi.org/10.1023/A:1006516607913Google Scholar
  9. P. Foralewski, H. Hudzik: Some basic properties of generalized Calderón-Lozanovskĭi spaces, Collect. Math. 48(4–6) (1997) 523–538.Google Scholar
  10. P. Foralewski, P. Kolwicz: Local uniform rotundity of Calderón-Lozanovskĭi spaces, J. Convex Analysis 14(2) (2007) 395–412.https://doi.org/10.68381/jca14026Google Scholar
  11. H. Hudzik, X. B. Liu, T. F. Wang: Points of monotonicity in Musielak-Orlicz function spaces endowed with the Luxemburg norm, Arch. Math. 82 (2004) 534-545.https://doi.org/10.1007/s00013-003-0440-xGoogle Scholar
  12. H. Hudzik, A. Narloch: Local monotonicity structure of Calderón-Lozanovskĭi spaces, Indag. Math., New Ser. 15(2) (2004) 245–255.https://doi.org/10.1016/S0019-3577(04)90017-1Google Scholar
  13. H. Hudzik, P. Kolwicz, A. Narloch: Local rotundity structure of Calderón-Lozanovskĭi spaces, Indag. Math., New Ser. 17(3) (2006) 373–395.https://doi.org/10.1016/S0019-3577(06)80039-XGoogle Scholar
  14. H. Hudzik, A. Kamińska, M. Mastyło: Geometric properties of some Calderón-Lozanovskĭi spaces and Orlicz-Lorentz spaces, Houston J. Math. 23(3) (1996) 639–663.Google Scholar
  15. H. Hudzik, M. Wisła: On extreme points of Orlicz spaces with Orlicz norm, Collect. Math. 44 (1993) 137–148.Google Scholar
  16. P. Kolwicz: Rotundity properties in Calderón-Lozanovskĭi spaces, Houston J. Math. 31(3) (2005) 883–912.Google Scholar
  17. P. Kolwicz: Kadec-Klee properties of Calderón-Lozanovskĭi function spaces, J. Funct. Spaces Appl. 2012 (2012), Article ID 314068, 21 pp.https://doi.org/10.1155/2012/314068Google Scholar
  18. P. Kolwicz: Kadec-Klee properties of Calderón-Lozanovskĭi sequence spaces, Collect. Math. 63(1) (2012) 45–58.https://doi.org/10.1007/s13348-010-0015-7Google Scholar
  19. P. Kolwicz, K. Leśnik: Topological and geometrical structure of Calderón-Lozanovskĭi construction, Math. Inequal. Appl. 13(1) (2010) 175–196.https://doi.org/10.7153/mia-13-14Google Scholar
  20. P. Kolwicz, K. Leśnik: Property β of Rolewicz and orthogonal convexities in Calderón-Lozanovskĭi spaces, Nonlinear Anal. 74 (2011) 4352–4368.https://doi.org/10.1016/j.na.2011.03.040Google Scholar
  21. P. Kolwicz, R. Płuciennik: On uniform rotundity in every direction in Calderón-Lozanovskĭi spaces, J. Convex Analysis 14(3) (2007) 621–645.https://doi.org/10.68381/jca14037Google Scholar
  22. P. Kolwicz, R. Płuciennik: Local ∆2 (x)-condition as a crucial tool for local structure of Calderón-Lozanovskĭi spaces, J. Math. Anal. Appl. 356 (2009) 605–614.https://doi.org/10.1016/j.jmaa.2009.03.030Google Scholar
  23. P. Kolwicz, R. Płuciennik: Points of upper local uniform monotonicity in Calderón-Lozanovskĭi spaces, J. Convex Analysis 17(1) (2010) 111–130.https://doi.org/10.68381/jca17010Google Scholar
  24. M. A. Krasnosielskii, Y. B. Rutickii: Convex Functions and Orlicz Spaces, Noordhoff, Groningen (1961) (in Russian).Google Scholar
  25. S. G. Krein, Y. I. Petunin, E. M. Semenov: Interpolation of Linear Operators, Nauka, Moscow (1978) (in Russian).Google Scholar
  26. J. Lindenstrauss, L. Tzafiri: Clasical Banach Spaces II, Springer, Berlin (1979).https://doi.org/10.1007/978-3-662-35347-9Google Scholar
  27. L. Maligranda: Orlicz Spaces and Interpolation, Seminar in Math. 5, Univ. Estadual de Campinas, Campinas (1989).Google Scholar
  28. M. M. Rao, Z. D. Ren: Theory of Orlicz Spaces, Marcel Dekker, New York (1991).Google Scholar
  29. A. Suarez-Granero, M. Wisła: Closedness of the set of extreme points in Orlicz spaces, Math. Nachr. 157 (1992) 319–334.https://doi.org/10.1002/mana.19921570125Google Scholar
  30. M. Wisła: A full description of extreme points in C(Ω, Lϕ (µ)), Proc. Amer. Math. Soc. 113(1) (1991) 193–200.https://doi.org/10.1090/S0002-9939-1991-1072351-6Google Scholar
  31. M. Wisła: Closedness of the set of extreme points in Orlicz spaces with Orlicz norm, Commentat. Math. 41 (2001) 221–239.Google Scholar
  32. A. C. Zaanen: Riesz Spaces II, North Holland, Amsterdam (1983).Google Scholar

Suggested citation

E. Kasior, M. Wisła. “Closedness of the Set of Extreme Points in Calderón-Lozanovskii Spaces.” Journal of Convex Analysis 21 (2014), No. 2, 401–413. https://doi.org/10.68381/jca21020

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Published by Heldermann Verlag, 2014. Rights now held by Banach Press.