Advertisement

Positivity

, Volume 18, Issue 4, pp 785–804 | Cite as

Domination and factorization theorems for positive strongly \(p\)-summing operators

  • D. AchourEmail author
  • A. Belacel
Article

Abstract

The aim of this work is to contribute to the theory of \((p,q)\)-summing operators. We focus on positive \((p,q)\)-summing operators, introduced by Blasco (Collect Math 37(1):13–22, 1986). We characterize their conjugates and provide new domination/factorization theorems for these classes. As an application, it is also shown that certain known results on \((p,q)\)-concave operators from Banach lattices can be lifted to a class of \((q,p)\)-convex operators.

Keywords

Banach lattice Positive \((p, q)\)-summing operators Pietsch-type theorem Strongly \((p, q)\)-summing operators Tensor norm 

Mathematics Subject Classification (2000)

46B42 46B45 46B28 47B10 

Notes

Acknowledgments

This work is a part of the doctoral thesis of A. Belacel. The authors wish to thank Professor E. A. Sánchez Pérez for his many useful suggestions concerning this paper.

References

  1. 1.
    Achour, D., Dahia, E., Rueda, P., Sanchez Pérez, E.A.: Factorization of strongly \((p;\sigma )\)-continuous multilinear operators. Linear Multilinear Algebra (2013). doi: 10.1080/03081087.2013.839677
  2. 2.
    Apiola, H.: Duality between spaces of \(p\)-summable sequences, \((p, q)\)-summing operators and characterizations of nuclearity. Math. Ann. 219, 53–64 (1976)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Blasco, O.: A class of operators from a Banach lattice into a Banach space. Collect. Math. 37(1), 13–22 (1986)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Blasco, O.: Positive \(p\)-summing operators on \(L^{p}\)-spaces. Proc. Am. Math. Soc. 100(2), 275–280 (1987)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Bu, Q., Buskes, G.: The Radon-Nikodym property for tensor products of Banach lattices. Positivity 10(2), 365–390 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Calabuig, J., Rodríguez, J., Sánchez Pérez, E.A.: Strongly embedded subspaces of \(p\)-convex Banach function spaces. Positivity 17, 775–791 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Chaney, J.: Banach lattices of compact maps. Math. Z. 129, 1–19 (1972)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Cohen, J.S.: Absolutely \(p\)-summing, \(p\)-nulear operators and thier conjugates. Math. Ann. 201, 177–200 (1973)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Curbera, G.P.: Banach space properties of \(L^{1}\) of a vector measure. Proc. Am. Math. Soc. 123, 3797–3806 (1995)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Defant, A., Floret, K.: Tensor norms and operator ideals. In: North-Holland Mathematics Studies, vol. 176. North-Holland, Amsterdam (1993)Google Scholar
  11. 11.
    Diestel, J., Jarchow, H., Tonge, A.: Absolutely Summing Operators. Cambridge University Press, Cambridge (1995)CrossRefzbMATHGoogle Scholar
  12. 12.
    Jarchow, H.: Locally Convex Spaces. B.G. Teubner, Stuttgart (1981)CrossRefzbMATHGoogle Scholar
  13. 13.
    Geĭler, V.A., Chuchaev, I.I.: The second conjugate of a summing operator. Izv. Vyssh. Uchebn. Zaved. Mat. 12, 17–22 (1982)Google Scholar
  14. 14.
    Krivine, J.L.: Théorèmes de factorisation dans les espaces réticulés. Séminaire Maurey-Schwartz (1973–74). Exposes 22–23, École Polytechnique, Paris (1974)Google Scholar
  15. 15.
    Labuschagne, C.C.A.: Riesz reasonable cross norms on tensor products of Banach lattices. Quaest. Math. 27, 243–266 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Labuschagne, C.C.A.: Preduals and nuclear operators associated with bounded, \(p\)-convex, \(p\)-concave and positive \(p\)-summing operators. Can. J. Math 59(3), 614–637 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces I and II. Springer, Berlin (1996)CrossRefGoogle Scholar
  18. 18.
    Meyer-Nieberg, P.: Banach Lattices. Springer, Berlin (1991)CrossRefzbMATHGoogle Scholar
  19. 19.
    Sánchez Pérez, E.A.: Compactness arguments for spaces of \(p\)-integrable functions with respect to a vector measure and factorization of operators through Lebesgue-Bochner spaces. Ill. J. Math. 45, 907–923 (2001)zbMATHGoogle Scholar
  20. 20.
    Schaefer, H.H.: Banach Lattices and Positive Operators. Springer, Berlin (1974)CrossRefzbMATHGoogle Scholar
  21. 21.
    Zhukova, O.I.: On modifications of the classes of \(p\)-nuclear, \(p\)-summing and \(p\)-integral operators. Sib. Math. J. 30(5), 894–907 (1998)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Laboratoire d’Analyse Fonctionnelle et Géométrie des EspacesUniversity of M’silaM’silaAlgeria

Personalised recommendations