Skip to main content
SpringerLink
Log in

Intrinsic characterizations of perturbation classes on some Banach spaces

  • Published:
Archiv der Mathematik Aims and scope Submit manuscript

Abstract

We investigate relationships between inessential operators and improjective operators acting between Banach spaces X and Y, emphasizing the case in which one of the spaces is a C(K) space. We show that they coincide in many cases, but they are different in the case X = Y = C(K 0), where K 0 is a compact space constructed by Koszmider.

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. P. Aiena, Fredholm and local spectral theory, with application to multipliers, Kluwer Acad. Publ., 2004.

  2. Aiena P., González M.: Essentially incomparable Banach spaces and Fredholm theory. Proc. R. Ir. Acad. 93A, 49–59 (1993)

    Google Scholar 

  3. Aiena P., González M.: On inessential and improjective operators. Studia Math. 131, 271–287 (1998)

    MATH  MathSciNet  Google Scholar 

  4. Aiena P., González M.: Examples of improjective operators. Math. Z. 233, 471–479 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. F. Albiac and N. Kalton, Topics in Banach space Theory, Graduate Texts in Math. 233, Springer-Verlag, 2006.

  6. Argyros S.A. et al.: Complementation and embeddings of c 0(I) in Banach spaces. Proc. London Math. Soc. 85, 742–768 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bourgain J.: H ∞ is a Grothendieck space. Studia Math. 75, 193–216 (1983)

    MATH  MathSciNet  Google Scholar 

  8. Dashiell F.: Nonweakly compact operators from order-Cauchy complete Banach lattices, with application to Baire classes. Trans. Amer. Math. Soc. 266, 397–413 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  9. J. Diestel, Geometry of Banach spaces—selected topics, Lecture Notes in Math. 485; Springer-Verlag, 1975.

  10. González M.: On essentially incomparable Banach spaces. Math. Z. 215, 621–629 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  11. M. González and A. Martínez-Abejón, Tauberian operators, Operator Theory: Advances and applications, 194. Birkhäuser Verlag, Basel, 2010.

  12. Gowers W.T., Maurey B.: Banach spaces with small spaces of operators. Math. Ann. 307, 543–568 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  13. P. Harmand, D. Werner, and W. Werner, M-ideals in Banach spaces and Banach algebras, Lecture Notes in Math. 1547; Springer-Verlag, 1993.

  14. Kalenda O.F.K.: Valdivia compact spaces in topology and Banach space theory. Extracta Math. 15, 1–85 (2000)

    MATH  MathSciNet  Google Scholar 

  15. Kisliakov S.V.: On the conditions of Dunford-Pettis, Pełczyński and Grothendieck. Dokl. Acad. Nauk. SSR 225, 152–155 (1975)

    Google Scholar 

  16. Kleinecke D.: Almost-finite, compact, and inessential operators. Proc. Amer. Math. Soc. 14, 863–868 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  17. Koszmider P.: Banach spaces of continuous functions with few operators. Math. Ann. 330, 151–183 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  18. Lotz H.P., Peck N.T., Porta H.: Semi-embeddings of Banach spaces. Proc. Edinburgh Math. Soc. 22, 233–240 (1979)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  20. Plebanek G.: A construction of a Banach space C(K) with few operators. Topology Appl. 143, 217–239 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  21. Pietsch A.: Inessential operators in Banach spaces. Integral Equations and Operator Theory 1, 589–591 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  22. Plichko A.M., Yost D.: Complemented and uncomplemented subspaces of Banach spaces. Extracta Math. 15, 335–371 (2000)

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  24. Tarafdar E.: Improjective operators and ideals in a category of Banach spaces. J. Austral. Math. Soc. 14, 274–292 (1972)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Metodi e Modelli Matematici, Facoltà di Ingegneria, Università di Palermo, Viale delle Scienze, 90128, Palermo, Italy

    Pietro Aiena

  2. Departamento de Matemáticas, Facultad de Ciencias, Universidad de Cantabria, 39071, Santander, Spain

    Manuel González

Authors
  1. Pietro Aiena
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Manuel González
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Manuel González.

Additional information

The first-named author is supported by Fondi ex-60 2008, Universitá di Palermo. The second-named author is supported in part by DGICYT (Spain), Grant MTM2007–67994.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Aiena, P., González, M. Intrinsic characterizations of perturbation classes on some Banach spaces. Arch. Math. 94, 373–381 (2010). https://doi.org/10.1007/s00013-010-0103-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00013-010-0103-7

Mathematics Subject Classification (2000)

Keywords

Search

Navigation