Advertisement

An isomorphic property in spaces of compact operators and some classes of operators on C(K,X)

  • I Ghenciu
Article
  • 1 Downloads

Abstract

Let \({K_{w^{*}}(X^{*},Y)}\) denote the set of all w*w continuous compact operators from X* to Y. We investigate whether the space \({K_{w^{*}}(X^{*},Y)}\) has property RDP p * (\({1\le p < \infty}\)) when X and \({Y}\) have the same property.

Suppose X and Y are Banach spaces, K is a compact Hausdorff space, \({\Sigma}\) is the \({\sigma}\) -algebra of Borel subsets of K, C(K,X) is the Banach space of all continuous X-valued functions (with the supremum norm), and \({T: C(K,X)\to Y}\) is a strongly bounded operator with representing measure \({m: \Sigma \to L(X,Y)}\).

We show that if T is a strongly bounded operator and \({\hat{T}: B(K, X) \to Y}\) is its extension, then T* is p-convergent if and only if \({\hat{T}^{*}}\) is p-convergent, for \({1\le p < \infty}\).

Key words and phrases

property \({RDP_{p}^{*}}\) space of compact operators p-convergent operator space of continuous functions 

Mathematics Subject Classification

primary 46B20 secondary 46B25 46B28 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    F. Albiac and N. J. Kalton, Topics in Banach Space Theory, Graduate Texts in Mathematics, 233, Springer (New York, 2006).Google Scholar
  2. 2.
    Kevin Andrews, Dunford–Pettis sets in the space of Bochner integrable functions, Math. Ann., 241 (1979), 35–41.Google Scholar
  3. 3.
    Bator E., Lewis P., Ochoa J.: Evaluation maps, restriction maps, and compactness. Colloq. Math., 78, 1–17 (1998)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bator E., Lewis P.: Operators having weakly precompact adjoints. Math. Nachr., 157, 99–103 (1992)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Batt J., Berg E.J.: Linear bounded transformations on the space of continuous functions. J. Funct. Anal., 4, 215–239 (1969)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bessaga C., Pelczynski A.: On bases and unconditional convergence of series in Banach spaces. Studia Math., 17, 151–174 (1958)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bombal F., Cembranos P.: Characterizations of some classes of operators on spaces of vector-valued continuous functions. Math. Proc. Camb. Philos. Soc., 97, 137–146 (1985)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Brooks J.K., Lewis P.: Linear operators and vector measures, Trans. Amer. Math. Soc., 192, 139–162 (1974)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Castillo J., Sanchez F.: Dunford–Pettis like properties of continuous vector function spaces. Rev. Mat. Univ. Complut. Madrid, 6, 43–59 (1993)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Carne T.K., Cole B., Gamelin T.W.: A uniform algebra of analytic functions on a Banach space. Trans. Amer. Math. Soc., 314, 639–659 (1989)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Cembranos P.: C(K, E) contains a complemented copy of c 0. Proc. Amer. Math. Soc., 91, 556–558 (1984)MathSciNetzbMATHGoogle Scholar
  12. 12.
    J. Diestel, H. Jarchow and A. Tonge, Absolutely Summing Operators, Cambridge Stud. Adv. Math., 43, Cambridge University Press (1995).Google Scholar
  13. 13.
    J. Diestel, Sequences and Series in Banach Spaces, Graduate Texts in Math., 92, Springer-Verlag (Berlin, 1984).Google Scholar
  14. 14.
    J. Diestel, A survey of results related to the Dunford–Pettis property, in: Proceedings of the Conference on Integration, Topology, and Geometry in Linear Spaces (Univ. North Carolina, Chapel Hill, N.C., 1979), Contemp. Math., 2, Amer. Math. Soc. (Providence, RI, 1980), pp. 15–60.Google Scholar
  15. 15.
    J. Diestel and J. J. Uhl, Jr., Vector Measures, Math. Surveys 15, Amer. Math. Soc. (providence, RI, 1977).Google Scholar
  16. 16.
    N. Dinculeanu, Vector Measures, Pergamon Press (1967).Google Scholar
  17. 17.
    Emmanuele G.: A dual characterization of Banach spaces not containing \({\ell^1}\). Bull. Polish Acad. Sci. Math., 34, 155–160 (1986)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Emmanuele G.: A remark on the containment of c 0 in spaces of compact operators. Math. Proc. Cambr. Philos. Soc., 111, 331–335 (1992)CrossRefGoogle Scholar
  19. 19.
    Emmanuele G.: Dominated operators on C[0,1] and the (CRP). Collect. Math., 41, 21–25 (1990)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Emmanuele G., John K.: Uncomplementability of spaces of compact operators in larger spaces of operators. Czechoslovak Math. J., 47, 19–31 (1997)MathSciNetCrossRefGoogle Scholar
  21. 21.
    I. Ghenciu and P. Lewis, Strongly bounded representing measures and convergence theorems, Glasgow Math. J.,  https://doi.org/10.1017/S0017089510000133, published online by Cambridge University Press (2010).
  22. 22.
    Ghenciu I.: On some classes of operators on C(K, X). Bull. Polish. Acad. Sci. Math., 63, 261–274 (2015)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Ghenciu I.: Weak precompactness and property (V *). Colloq. Math., 38, 255–269 (2014)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Ghenciu I., Lewis P.: The embeddability of c 0 in spaces of operators. Bull. Polish. Acad. Sci. Math., 56, 239–256 (2008)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Ghenciu I., Lewis P.: Almost weakly compact operators. Bull. Polish. Acad. Sci. Math., 54, 237–256 (2006)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Ghenciu I., Lewis P.: The Dunford–Pettis property, the Gelfand-Phillips property, and L-sets. Colloq. Math., 106, 311–324 (2006)MathSciNetCrossRefGoogle Scholar
  27. 27.
    I. Ghenciu, The p-Gelfand–Phillips property in spaces of operators and Dunford–Pettis like sets, Acta Math. Hungar., to appear.Google Scholar
  28. 28.
    Kalton N.: Spaces of compact operators. Math. Ann., 208, 267–278 (1974)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Lust F.: Produits tensoriels injectifs d’espaces de Sidon. Colloq. Math., 32, 285–289 (1975)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Pełczyński A.: Banach spaces on which every unconditionally converging operator is weakly compact. Bull. Acad. Polon. Sci. Math. Astronom. Phys., 10, 641–648 (1962)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Pełczyński A., Semadeni Z.: Spaces of continuous functions. III. Studia Math., 18, 211–222 (1959)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Pitt H.R.: A note on bilinear forms. J. London Math. Soc., 11, 174–180 (1936)MathSciNetCrossRefGoogle Scholar
  33. 33.
    G. Pisier, Factorization of Linear Operators and Geometry of Banach Spaces, CBMS Regional Conf. Series in Math., 60, Amer. Math. Soc. (Providence, RI, 1986).Google Scholar
  34. 34.
    Rosenthal H.P.: On quasi-complemented subspaces of Banach spaces, with an appendix on compactness of operators from \({L^p(\mu)}\) to \({L^r(\nu)}\). J. Funct. Anal., 4, 176–214 (1969)CrossRefGoogle Scholar
  35. 35.
    Rosenthal H.: Pointwise compact subsets of the first Baire class. Amer. J. Math., 99, 362–377 (1977)MathSciNetCrossRefGoogle Scholar
  36. 36.
    W. Ruess, Duality and geometry of spaces of compact operators, in: Functional Analysis: Surveys and Recent Results. III, Proc. 3rd Paderborn Conference, 1983, North-Holland Math. Studies no. 90 (1984), pp. 59–78.Google Scholar
  37. 37.
    R. A. Ryan, Introduction to Tensor Products of Banach Spaces, Springer (London, 2002).Google Scholar
  38. 38.
    Ryan R.: The Dunford–Pettis property and projective tensor products. Bull. Polish Acad. Sci. Math., 35, 785–792 (1987)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Z. Semadeni, Banach Spaces of Continuous Functions, PWN (Warsaw, 1971).Google Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  1. 1.Mathematics DepartmentUniversity of Wisconsin–River FallsWisconsinU.S.A.

Personalised recommendations