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Strongly limited (Dunford–Pettis) completely continuous subspaces of operator ideals

  • Halimeh ArdakaniEmail author
  • Manijeh Salimi
  • Seyed Mohammad Moshtaghioun
Original Paper

Abstract

By introducing the concepts of strongly limited completely continuous and strongly Dunford–Pettis completely continuous subspaces of operator ideals, it will be given some characterizations of these concepts in terms of lcc ness and DPcc ness of all their evaluation operators related to that subspace. In particular, when \(X^*\) or Y has the Gelfand–Phillips (GP) property, we give a characterization of GP property of a closed subspace \({\mathcal {M}}\) of compact operators K(XY) in terms of strong limited complete continuity of \({\mathcal {M}}\). Also it is shown that, each operator ideal \({\mathcal {U}}(X, Y )\) is strongly limited completely continuous, iff, \(X^*\) and Y have the GP property.

Keywords

Gelfand–Phillips property Completely continuous algebra Strongly completely continuous algebra Limited completely continuous operator 

Mathematics Subject Classification

47L05 47L20 46B28 46B99 

References

  1. 1.
    Aliprantis, C.D., Burkishaw, O.: Positive Operators. Academic Press, New York (1978)Google Scholar
  2. 2.
    Anselone, P.M.: Compactness properties of sets of operators and their adjoints. Math. Z. 113, 233–236 (1970)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ardakani, H., Salimi, M.: L-Dunford–Pettis and almost L-Dunford–Pettis sets in dual Bnach lattices. Int. J. Anal. Appl. 16, 149–161 (2018)zbMATHGoogle Scholar
  4. 4.
    Borwein, J., Fabian, M., Vanderwerff, J.: Characterizations of Banach spaces via convex and other locally Lipschitz functions. Acta Math. Vietnam 1, 53–69 (1997)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Brown, S.W.: Weak sequential convergence in the dual of an algebra of compact operators. J. Oper. Theory 33, 33–42 (1995)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Defant, A., Floret, K.: Tensor Norms and Operator Ideals. Mathematical Studies, vol. 179. North-Holland, Amsterdam (1993)zbMATHGoogle Scholar
  7. 7.
    Diestel, J.: Sequences and Series in Banach Spaces. Graduate Texts in Mathematics, vol. 92. Springer, Berlin (1984)CrossRefGoogle Scholar
  8. 8.
    Drewnowski, L.: On Banach spaces with the Gelfand–Phillips property. Math. Z. 193, 405–411 (1986)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dunford, N., Schwartz, J.T.: Linear Operators, I: General Theory, Pure and Applied Mathematics, vol. 7. Interscience, New York (1958)zbMATHGoogle Scholar
  10. 10.
    Emmanuele, G.: A dual characterization of Banach spaces not containing \(\ell _1\). Bull. Pol. Acad. Sci. Math. 34, 155–160 (1986)zbMATHGoogle Scholar
  11. 11.
    Emmanuele, G.: Banach spaces in which Dunford–Pettis sets are relatively compact. Arch. Math. 58, 477–485 (1992)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Emmanuele, G.: On Banach spaces with the Gelfand–Phillips property, III. J. Math. Pures Appl. 72, 327–333 (1993)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Moshtaghioun, S.M., Zafarani, J.: Completely continuous subspaces of operator ideals. Taiwan. J. Math. 10, 691–698 (2006)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Moshtaghioun, S.M.: Weakly completely continuous subspaces of operator ideals. Taiwan. J. Math. 2, 523–530 (2007)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Musial, K.: The weak Radon–Nikodym property in Banach spaces. Stud. Math. 64, 151–173 (1979)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Palmer, T.W.: Totally bounded sets of precompact linear operators. Proc. Am. Math. Soc. 20, 101–106 (1969)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Salimi, M., Moshtaghioun, S.M.: A new class of Banach spaces and its relation with some geometric properties of Bancah spaces. Abstr. Appl. Anal. 1, 4 (2012)zbMATHGoogle Scholar
  18. 18.
    Salimi, M., Moshtaghioun, S.M.: The Gelfand–Phillips property in closed subspaces of some operator spaces. Banach J. Math. Anal. 2, 84–92 (2011)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Saksman, E., Tylli, H.O.: Structure of subspaces of the compact operators having the Dunford–Pettis property. Math. Z. 232, 411–435 (1999)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Sclumprecht, T.: Limited sets in injective tensor products. In: Odell, E., Rosenthal, H. (eds.) Lecture Notes in Mathematics, vol. 1970, pp. 133–158. Springer, Berlin (1991)Google Scholar
  21. 21.
    \(\ddot{U}\)lger, A.: Subspaces and subalgebras of K(H) whose duals have the Schur property. J. Oper. Theory 37, 371–378 (1997)Google Scholar
  22. 22.
    Wen, Y., Chen, J.: Characterizations of Banach spaces with relatively compact Dunford–Pettis sets. Adv. Math. 45, 122–132 (2016)MathSciNetzbMATHGoogle Scholar

Copyright information

© Tusi Mathematical Research Group (TMRG) 2019

Authors and Affiliations

  • Halimeh Ardakani
    • 1
    Email author
  • Manijeh Salimi
    • 2
  • Seyed Mohammad Moshtaghioun
    • 3
  1. 1.Department of MathematicsPayame Noor UniversityTehranIran
  2. 2.Department of MathematicsFarhangian UniversityTehranIran
  3. 3.Department of MathematicsYazd UniversityYazdIran

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