Advertisement

Positivity

, Volume 22, Issue 5, pp 1419–1431 | Cite as

On the class of disjoint limited completely continuous operators

  • Jawad H’michane
  • Noufissa Hafidi
  • Larbi Zraoula
Article
  • 35 Downloads

Abstract

We introduce and study new class of sets (almost L-limited sets). Also, we introduce new concept of property in Banach lattice (almost Gelfand–Phillips property) and we characterize this property using almost L-limited sets. On the other hand, we introduce the class of disjoint limited completely continuous operators which is a largest class than that of limited completely continuous operators, we characterize this class of operators and we study some of its properties.

Keywords

Limited set Almost L-limited set Gelfand–Phillips property Almost Gelfand–Phillips property Disjoint limited completely continuous operators Order continuous norm 

Mathematics Subject Classification

46B42 47B60 47B65 

References

  1. 1.
    Aliprantis, C.D., Burkinshaw, O.: Positive Operators. Reprint of the 1985 original. Springer, Dordrecht (2006)CrossRefGoogle Scholar
  2. 2.
    Aqzzouz, B., Bouras, K.: (L) sets and almost (L) sets in Banach lattices. Quaest. Math. 36, 107–118 (2013)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Aqzzouz, B., Bouras, K.: Dunford–Pettis sets in Banach lattices. Acta Math. Univ. Comen. LXXXI(2), 185–196 (2012)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bukhvalov, A.V.: Locally convex spaces that are generated by weakly compact sets, Vestnik Leningrad. Univ. No. 7 Mat. Meh. Astronorm. Vyp. 2, 11–17, 160 (1973)Google Scholar
  5. 5.
    Chen, J.X., Chen, Z.L., Ji, G.X.: Almost limited sets in Banach lattices. J. Math. Anal. Appl. 412, 547–553 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Drewnowski, L.: On Banach spaces with the Gelfand Phillips property. Math. Z. 193, 405–411 (1986)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Emmanuele, G.: A dual characterization of Banach spaces not containing \(\ell ^1\). Bull. Polish. Acad. Sci. Math. 34, 155–160 (1986)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Ardakani, H., Moshtaghioun, S.M., Mosadegh, Modarres S.S.M., Salimi, M.: The strong Gelfand–Phillips property in Banach lattices. Banach J. Math. Anal. 10(1), 15–26 (2016)MathSciNetCrossRefGoogle Scholar
  9. 9.
    H’michane, J., Elfahri, K.: On the domination of limited and order Dunford–Pettis operators. Ann. Math. Qubec 39, 169–176 (2015).  https://doi.org/10.1007/s40316-015-0036-4 MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Machrafi, N., Elbour, A., Moussa, M.: Some characterizations of almost limited sets and applications. arXiv:1312.2770 (2014)
  11. 11.
    Salimi, M., Moshtaghioun, S.M.: A new class of Banach spaces and its relation with some geometric properties of Banach spaces. J. Abstr. Appl. Anal. (2012).  https://doi.org/10.1155/2012/212957 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Salimi, M., Moshtaghioun, S.M.: The Gelfand–Phillips property in closed subspaces of some operator spaces. Banach J. Math. Anal. 5(2), 84–92 (2011)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Wickstead, A.W.: Converses for the Dodds–Fremlin and Kalton–Saab theorems. Math. Proc. Camb. Philos. Soc. 120, 175–179 (1996)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Wnuk, W.: Banach lattices with the weak Dunford–Pettis property. Atti Sem. Mat. Fis. Univ. Modena 42(1), 227–236 (1994)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Jawad H’michane
    • 1
  • Noufissa Hafidi
    • 1
  • Larbi Zraoula
    • 2
  1. 1.Département de Mathématiques, Faculté des SciencesUniversité Moulay IsmailZitoune, MeknesMorocco
  2. 2.Centre régional des métiers de l’éducation et de la formation Rabat-Salé-Kénitra Annexe Kénitra, Laboratoire de recherche en enseignement, apprentissage, mathématiques et applicationsCRMEF-KénitraKénitraMorocco

Personalised recommendations