Skip to main content
SpringerLink
Log in

Injective Dual Banach Spaces and Operator Ideals

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

We give characterizations of Banach spaces with \(\lambda \)-injective biduals in terms of operators with an integral representation. These results may be thought of as a certain refinement of Lindenstrauss’ extension theorems from his 1964 memoir. We also obtain dual results characterizing Banach spaces with \(\lambda \)-injective duals in terms of factorization of operators through \(L_1(\mu )\) or \(\ell _1\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Albiac, F., Kalton, N.J.: Topics in Banach Space Theory, Graduate Texts in Mathematics, vol. 233. Springer, Berlin (2006)

    MATH  Google Scholar 

  2. Bourgain, J., Delbaen, F.: A class of special \({\mathscr {L}}_\infty \) spaces. Acta Math. 145, 155–176 (1980)

    Article  MathSciNet  Google Scholar 

  3. Castillo, J.M.F., Moreno, Y., Suárez, J.: On Lindenstrauss-Pełczyński spaces. Stud. Math. 174, 213–231 (2006)

    Article  Google Scholar 

  4. Cilia, R., Gutiérrez, J.M.: Left \(\ell _1\)-factorable polynomials. Glasgow Math. J. 51, 631–649 (2009)

    Article  MathSciNet  Google Scholar 

  5. Cilia, R., Gutiérrez, J.M.: Operators with an integral representation. Proc. Am. Math. Soc. 144, 5275–5290 (2016)

    Article  MathSciNet  Google Scholar 

  6. Cilia, R., Gutiérrez, J.M.: Corrigendum to “Operators with an integral representation”. Proc. Am. Math. Soc. 148, 4117–4118 (2020)

    Article  MathSciNet  Google Scholar 

  7. Davis, W.J., Figiel, T., Johnson, W.B., Pełczyński, A.: Factoring weakly compact operators. J. Funct. Anal. 17, 311–327 (1974)

    Article  MathSciNet  Google Scholar 

  8. Defant, A., Floret, K.: Tensor Norms and Operator Ideals, Mathematical Studies, vol. 176. North-Holland, Amsterdam (1993)

    MATH  Google Scholar 

  9. Diestel, J., Jarchow, H., Tonge, A.: Absolutely Summing Operators, Cambridge Studies in Advanced Mathematics, vol. 43. Cambridge University Press, Cambridge (1995)

    Book  Google Scholar 

  10. Diestel, J., Uhl Jr., J.J.: Vector Measures, Mathematical Surveys Monographs, vol. 15. American Mathematical Society, Providence (1977)

    Google Scholar 

  11. Grothendieck, A.: Sur les applications linéaires faiblement compactes d’espaces du type \(C(K)\). Can. J. Math. 5, 129–173 (1953)

    Article  Google Scholar 

  12. Haydon, R.: On dual \(L_1\)-spaces and injective bidual Banach spaces. Isr. J. Math. 31, 142–152 (1978)

    Article  Google Scholar 

  13. Johnson, W.B.: Factoring compact operators. Isr. J. Math. 9, 337–345 (1971)

    Article  MathSciNet  Google Scholar 

  14. Johnson, W.B., Rosenthal, H.P., Zippin, M.: On bases, finite dimensional decompositions and weaker structures in Banach spaces. Isr. J. Math. 9, 488–506 (1971)

    Article  MathSciNet  Google Scholar 

  15. Kelley, J.L.: General Topology, Graduate Texts in Mathematics, vol. 27. Springer, New York (1975)

    Google Scholar 

  16. Lima, Å., Nygaard, O., Oja, E.: Isometric factorization of weakly compact operators and the approximation property. Isr. J. Math. 119, 325–348 (2000)

    Article  MathSciNet  Google Scholar 

  17. Lindenstrauss, J.: Extensions of Compact Operators, Memoirs of the American Mathematical Society, vol. 48. American Mathematical Society, Providence (1964)

    Google Scholar 

  18. Lindenstrauss, J., Pełczyński, A.: Absolutely summing operators in \({{\mathscr {L}}}_p\)-spaces and their applications. Stud. Math. 29, 275–326 (1968)

    Article  Google Scholar 

  19. Lindenstrauss, J., Rosenthal, H.P.: The \({{\mathscr {L}}}_p\)-spaces. Isr. J. Math. 7, 325–349 (1969)

    Article  Google Scholar 

  20. Megginson, R.E.: An Introduction to Banach Space Theory, Graduate Texts in Mathematics, vol. 183. Springer, Berlin (1998)

    Book  Google Scholar 

  21. Pełczyński, A.: Banach spaces on which every unconditionally converging operator is weakly compact. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 10, 641–648 (1962)

    MathSciNet  MATH  Google Scholar 

  22. Rosenthal, H.P.: A characterization of Banach spaces containing \(\ell _1\). Proc. Natl. Acad. Sci. 71, 241–243 (1974)

    Article  Google Scholar 

  23. Rosenthal, H.P.: The Banach spaces \(C(K)\). In: Johnson, W.B., Lindenstrauss, J. (eds.) Handbook of the Geometry of Banach Spaces. North-Holland, vol. 2, pp. 1547–1602. Elsevier, Amsterdam (2003)

    Chapter  Google Scholar 

  24. Zippin, M.: Extension of bounded linear operators. In: Johnson, W.B., Lindenstrauss, J. (eds.) Handbook of the Geometry of Banach Spaces, vol. 2, pp. 1703–1741. Amsterdam, North-Holland (2003)

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Facoltà di Scienze, Università di Catania, Viale Andrea Doria 6, 95125, Catania, Italy

    Raffaella Cilia

  2. Departamento de Matemática Aplicada a la Ingeniería Industrial, Escuela Técnica Superior de Ingenieros Industriales, Universidad Politécnica de Madrid, C. José Gutiérrez Abascal 2, 28006, Madrid, Spain

    Joaquín M. Gutiérrez

Authors
  1. Raffaella Cilia
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Joaquín M. Gutiérrez
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Joaquín M. Gutiérrez.

Additional information

To the memory of our beloved and admired Professor Eve Oja.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Raffaella Cilia was supported in part by G.N.A.M.P.A. (Italy). Raffaella Cilia, Joaquín M. Gutiérrez were supported in part by Secretaría de Estado de Investigación, Desarrollo e Innovación, PGC2018-097286-B-I00 (Spain).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cilia, R., Gutiérrez, J.M. Injective Dual Banach Spaces and Operator Ideals. Mediterr. J. Math. 18, 12 (2021). https://doi.org/10.1007/s00009-020-01654-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-020-01654-9

Keywords

Mathematics Subject Classification

Search

Navigation