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On the Banach spaces with the property (V*) of Pelczynski

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Summary

We consider the Banach spaces with the property (V*) of Pelczynski giving a sufficient condition for a Banach space to have this property as well as a characterization of Banach lattices with the same property. Several other results are given which are concerning relationships among that property and other famous isomorphic properties of Banach spaces. Also a characterization of Banach spaces with property (V*) using Schauder decompositions is given. Some result concerning lifting of that property from a Banach space E to L1(Μ, E) is presented, too.

References

  1. [1]

    C. Bessaga -A. Pelczynski,On bases and unconditional convergence in Banach spaces, Studia Math.,17 (1958), pp. 151–164.

  2. [2]

    J. Bourgain,An averaging result for l 1 sequences and application to weakly conditionally compact set in L x 1 , Israel J. Math.,32 (1979), pp. 289–298.

  3. [3]

    J. Bourgain -F. Delbaen, A special class ofL , Acta Math.,145 (1980), pp. 155–176.

  4. [4]

    J.Diestel - J. J.Uhl jr.,Vector measures, Math. Surveys,15 AMS (1977).

  5. [5]

    J.Diestel,Sequences and series in Banach spaces, Graduate Texts in Math., Springer-Verlag, 1984.

  6. [6]

    N.Dunford - J. T.Schwartz,Linear Operators, I, Interscience, 1958.

  7. [7]

    G. A. Edgar,An ordering for the Banach spaces, Pacific J. Math.,108 (1983), pp. 83–98.

  8. [8]

    G.Emmanuele,On the Dieudonné property, Comm. Math., to appear.

  9. [9]

    G.Emmanuele,On weak compactness in Banach spaces of E-valued measures and E-valued Bochner integrable functions, when E has an unconditional Schauder decomposition, Revue Roumanie Math. Pures Appl., to appear.

  10. [10]

    J. Figiel -N. Ghoussoub -W. B. Johnson,On the structure of nonweakly compact operators on Banach lattices, Math. Annalen,257 (1981), pp. 317–334.

  11. [11]

    C.Fierro Bello,On weakly compact and unconditionally converging operators in spaces of vector valued continuous functions, preprint.

  12. [12]

    J. L. B. Gamlen,On a theorem of Pelczynski, Proc. Amer. Math. Soc.,44 (1974), pp. 283–285.

  13. [13]

    G. Godefroy -P. Saab,Lès propriétés (V)et (V*)de A. Pelczynski: quelques nouveaux exemples, Compte Rendu Acad. Sci. Paris,303 (1986), pp. 503–506.

  14. [14]

    A. Grothendieck,Sur les applications linéaires faiblement compactes d'espaces du type C(K), Canad. J. Math.,5 (1953), pp. 129–173.

  15. [15]

    T.Leavelle,The reciprocal Dunford-Pettis property, Annali Mat. Pura Appl., to appear.

  16. [16]

    C. P. Niculescu,Weak compactness in Banach lattices, J. Operator Theory,9 (1981), pp. 217–231.

  17. [17]

    A. Pelczynski,Banach spaces on which every unconditionally converging operator is weakly compact, Bull. Acad. Pol. Sci.,10 (1962), pp. 641–648.

  18. [18]

    A.Pelczynski,Banach spaces of analytic functions and absolutely summing operators, CBMS 30, AMS 1977.

  19. [19]

    H. P. Rosenthal,On relatively disjoint families of measures with some applications to Banach space theory, Studia Math.,37 (1970), pp. 13–36.

  20. [20]

    E. Saab -P. Saab,A stability property of a class of Banach spaces not containing a complemented copy of l 1, Proc. Amer. Math. Soc.,84 (1982), pp. 44–46.

  21. [21]

    E. Saab -P. Saab,On Pelczynski's properties (V)and (V*), Pacific J. Math.,125 (1986), pp. 205–210.

  22. [22]

    I.Singer,Bases in Banach spaces, II, Springer-Verlag, 1981.

  23. [23]

    M. Talagrand,Weak Cauchy sequences in L 1 (E), Amer. J. Math.,106 (1984), pp. 703–724.

  24. [24]

    M. Talagrand,Quand l'espace des measures a variation bornée est il faiblement séquentiellement complete, Proc. Amer. Math. Soc.,90 (1984), pp. 285–288.

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Work performed under the auspices of G.N.A.F.A. of C.N.R. and partially supported by M.P.I. of Italy (40%).

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Emmanuele, G. On the Banach spaces with the property (V*) of Pelczynski. Annali di Matematica pura ed applicata 152, 171–181 (1988). https://doi.org/10.1007/BF01766147

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Keywords

  • Banach Space
  • Banach Lattice
  • Isomorphic Property