Advertisement

Positivity

, Volume 20, Issue 1, pp 155–169 | Cite as

Tensor product representation of Köthe-Bochner spaces and their dual spaces

  • J. M. Calabuig
  • E. Jiménez Férnandez
  • M. A. Juan
  • E. A. Sánchez PérezEmail author
Article
  • 174 Downloads

Abstract

We provide a tensor product representation of Köthe-Bochner function spaces of vector valued integrable functions. As an application, we show that the dual space of a Köthe-Bochner function space can be understood as a space of operators satisfying a certain extension property. We apply our results in order to give an alternate representation of the dual of the Bochner spaces of p-integrable functions and to analyze some properties of the natural norms \(\Delta _p\) that are defined on the associated tensor products.

Keywords

Vector measure Köthe-Bochner space Tensor product Dual space 

Mathematics Subject Classification

46E30 46E40 46B28 46G10 46B42 

References

  1. 1.
    Bochner, S.: Integration von Funktionen, deren Werte die Elemente eines Vectorraumes sind. Fundamenta Mathematicae 20, 262–276 (1933)Google Scholar
  2. 2.
    Calabuig, J.M., Delgado, O., Juan, M.A., Sánchez, E.A.: Pérez, On the Banach lattice structure of \(L^1_w\) of a vector measure on a \(\delta \)-ring. Collect. Math. 65, 6567–85 (2014)CrossRefGoogle Scholar
  3. 3.
    Calabuig, J.M., Delgado, O., Sánchez Pérez, E.A.: Factorizing operators on Banach function spaces through spaces of multiplication operators. J. Math. Anal. Appl. 364(1), 88–103 (2010)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Calabuig, J.M., Gregori, P., Sánchez, E.A.: Pérez, Radon-Nikodým derivatives for vector measures belonging to Köthe function spaces. J. Math. Anal. Appl. 348, 469–479 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Cerdà, J., Hudzik, H., Mastyło, M.: Geometric properties of Köthe-Bochner spaces. Math. Proc. Cambridge Philos. Soc. 120(3), 521–533 (1996)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Chakraborty, N.D., Basu, S.: Spaces of p-tensor integrable functions and related Banach space properties. Real Anal. Exchange 34, 87–104 (2008)Google Scholar
  7. 7.
    Chakraborty, N.D., Basu, S.: Integration of vector-valued functions with respect to vector measures defined on \(\delta \)-rings. Ill. J. Math. 55(2), 495–508 (2011)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Defant, A., Floret, K.: Tensor norms and operator ideals. North-Holland, Amsterdam (1993)zbMATHGoogle Scholar
  9. 9.
    Delgado, O., Juan, M.A.: Representation of Banach lattices as \(L^{1}_{w}\) spaces of a vector measure defined on a \(\delta -\)ring. Bull. Belgian Math. Soc. 19, 239–256 (2012)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Diestel, J., Uhl, J.J.: Vector measures. Am. Math. Soc, Providence (1977)CrossRefzbMATHGoogle Scholar
  11. 11.
    Dobrakov, I.: On integration in Banach spaces, VII. Czechoslovak Math. J. 38, 434–449 (1988)MathSciNetGoogle Scholar
  12. 12.
    García-Raffi, L.M., Jefferies, B.: An application of bilinear integration to quantum scattering. J. Math. Anal. Appl. 415, 394–421 (2014)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Gregori Huerta, P.: Espacios de medidas vectoriales. Thesis, Universidad de Valencia, ISBN:8437060591 (2005)Google Scholar
  14. 14.
    Jefferies, B., Okada, S.: Bilinear integration in tensor products. Rocky Mt. J. Math. 28, 517–545 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Lewis, D.R.: On integrability and summability in vector spaces. Ill. J. Math. 16, 294–307 (1972)zbMATHGoogle Scholar
  16. 16.
    Lin, P.-K.: Köthe-Bochner function spaces. Birkhauser, Boston (2004)CrossRefzbMATHGoogle Scholar
  17. 17.
    Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces II. Springer, Berlin (1979)CrossRefzbMATHGoogle Scholar
  18. 18.
    Okada, S., Ricker, W.J., Sánchez Pérez, E.A.: Optimal Domains and integral extensions of operators acting in function spaces. Operator Theory Advances and Applications, vol. 180. Birkhäuser, Basel (2008)Google Scholar
  19. 19.
    Pallu de La Barriére, R.: Integration of vector functions with respect to vector measures. Studia Univ. Babes-Bolyai Math. 43, 55–93 (1998)MathSciNetGoogle Scholar
  20. 20.
    Rodríguez, J.: On integration of vector functions with respect to vector measures. Czechoslovak Math. J. 56, 805–825 (2006)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  • J. M. Calabuig
    • 1
  • E. Jiménez Férnandez
    • 2
  • M. A. Juan
    • 3
  • E. A. Sánchez Pérez
    • 1
    Email author
  1. 1.Instituto Universitario de Matemática Pura y AplicadaUniversitat Politècnica de ValènciaValenciaSpain
  2. 2.Departamento de Economía AplicadaUniversitat Jaume ICastellónSpain
  3. 3.Universidad Católica de ValenciaValenciaSpain

Personalised recommendations