Skip to main content
SpringerLink
Log in

Linear maps that preserve semi-Fredholm operators acting on Banach spaces

  • Published:
Acta Scientiarum Mathematicarum Aims and scope Submit manuscript

Abstract

We consider the linear maps φ: B(X) → B(Y) that preserve the semi-Fredholm operators in both directions or the essential spectrum of an operator, where B(Y) is the algebra of all bounded linear operators on an infinite-dimensional Banach space X. We describe some known results in the Hilbert space case, provide some basic results and examples in the general case, and state several open problems.

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

    MathSciNet  MATH  Google Scholar 

  2. B. Aupetit, A primer on spectral theory, Springer-Verlag, New York, 1991.

    Book  Google Scholar 

  3. B. Aupetit, Sur les transformations qui conservent le spectre, Banach Algebra’97, Walter de Gruyter, Berlin, 1998, 55–78.

    Google Scholar 

  4. B. Aupetit, Spectrum-preserving linear mappings between Banach algebras or Jordan-Banach algebras, J. London Math. Soc., 62 (2000), 917–924.

    Article  MathSciNet  Google Scholar 

  5. M. Bendaoud, A. Bourhim, M. Burgos and M. Sarih, Linear maps preserving Fredholm and Atkinson elements of C*-algebras, Linear and Multilinear Algebra, 57 (2009), 823–838.

    Article  MathSciNet  Google Scholar 

  6. M. Brešar and P. Šemrl, Linear maps preserving the spectral radius, J. Funct. Anal., 142 (1996), 360–368.

    Article  MathSciNet  Google Scholar 

  7. M. Brešar and P. Šemrl, Invertibility preserving maps preserve idempotents, Michigan Math. J., 45 (1998), 483–488.

    Article  MathSciNet  Google Scholar 

  8. S. R. Caradus, W. E. Pfaffenberger and B. Yood, Calkin Algebras and Algebras of Operators on Banach Spaces, Marcel Dekker, New York, 1974.

    MATH  Google Scholar 

  9. J. Dieudonné, Sur une généralisation du groupe orthogonal à quatre variables, Arch. Math (Basel), 1 (1949), 282–287.

    Article  Google Scholar 

  10. R. G. Douglas, Banach Algebra Techniques in Operator Theory, Academic Press, 1972.

    MATH  Google Scholar 

  11. I. Farah, All automorphisms of the Calkin algebra are inner, Ann. of Math. (2), 173 (2011), 619–661.

    Article  MathSciNet  Google Scholar 

  12. M. González, The perturbation classes problem in Fredholm theory, J. Funct. Anal., 200 (2003), 65–70.

    Article  MathSciNet  Google Scholar 

  13. M. González and J. M. Herrera, Spaces on which the essential spectrum of all the operators is finite, J. Operator Theory, 53 (2005), 303–314.

    MathSciNet  MATH  Google Scholar 

  14. M. González, A. Martínez-Abejón and M. Salas-Brown, Perturbation classes for semi-Fredholm operators on subprojective and superprojective spaces, Ann. Acad. Sci. Fennicae Math., 36 (2011), 481–491.

    Article  MathSciNet  Google Scholar 

  15. M. González and J. Pello, Superprojective Banach spaces, J. Math. Anal. Appl., 437 (2016), 1140–1151.

    Article  MathSciNet  Google Scholar 

  16. A. Guterman, C. -K. Li and P. Šemrl, Some general techniques on linear preserver problems, Linear Algebra Appl., 315 (2000), 61–81.

    Article  MathSciNet  Google Scholar 

  17. J. Hou and J. Cui, Linear maps preserving essential spectral functions and closedness of operator ranges, Bull. London Math. Soc., 39 (2007), 575–582.

    Article  MathSciNet  Google Scholar 

  18. A. Jafarian and A. R. Sourour, Spectrum preserving linear maps, J. Funct. Anal., 66 (1986), 255–261.

    Article  MathSciNet  Google Scholar 

  19. I. Kaplansky, Algebraic and analytic aspects of operator algebras, Amer. Math. Soc., Providence, 1970.

    Book  Google Scholar 

  20. A. Lebow and M. Schechter, Semigroups of operators and measures of noncompactness, J. Funct. Anal., 7 (1971), 1–26.

    Article  MathSciNet  Google Scholar 

  21. M. Marcus and R. Purves, Linear transformations on algebras of matrices: The invariance of the elementary symmetric functions, Canad. J. Math., 11 (1959), 383–396.

    Article  MathSciNet  Google Scholar 

  22. M. Mbekhta, Linear maps preserving the set of Fredholm operators, Proc. Amer. Math. Soc., 135 (2007), 3613–3619.

    Article  MathSciNet  Google Scholar 

  23. M. Mbekhta, A survey on linear (additive) preserver problems, Advanced courses of mathematical analysis IV, World Sci. Publ., Hackensack, 2012, 174–195.

    MATH  Google Scholar 

  24. M. Mbekhta, L. Rodman and P. Šemrl, Linear maps preserving generalized invertibility, Integral Equations Operator Theory, 55 (2006), 93–109.

    Article  MathSciNet  Google Scholar 

  25. M. Mbekhta and P. Šemrl, Linear maps preserving semi-Fredholm operators and generalized invertibility, Linear and Multilinear Algebra, 57 (2009), 55–64.

    Article  MathSciNet  Google Scholar 

  26. L. Molnár, Selected Preserver Problems on Algebraic Structures of Linear Operators and on Function Spaces, Lecture Notes in Math. 1895, Springer-Verlag, Berlin, 2007.

    MATH  Google Scholar 

  27. V. Müller, Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras, Birkhäuser, Basel, 2003.

    Book  Google Scholar 

  28. T. Oikhberg and E. Spinu, Subprojective Banach spaces, J. Math. Anal. Appl., 424 (2015), 613–635.

    Article  MathSciNet  Google Scholar 

  29. M. Omladic and P. Šemrl, Additive mappings preserving operators of rank one, Linear Algebra and Applications, 182 (1993), 239–256.

    Article  MathSciNet  Google Scholar 

  30. N. C. Phillips and N. Weaver, The Calkin algebra has outer automorphisms, Duke Math. J., 139 (2007), 185–202.

    Article  MathSciNet  Google Scholar 

  31. S. Pierce et al., A survey of linear preserver problems, Linear and Multilinear Algebra, 33 (1992), 1–192.

    Article  MathSciNet  Google Scholar 

  32. A. R. Sourour, Invertibility preserving linear maps on −(X), Trans. Amer. Math. Soc., 348 (1996), 13–30.

    Article  MathSciNet  Google Scholar 

  33. R. J. Whitley, Strictly singular operators and their conjugates, Trans. Amer. Math. Soc., 113 (1964), 252–261.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemáticas, Universidad de Cantabria, Avenida de los Castros s/n, 39071, Santander, Spain

    Manuel González

  2. UFR de Mathématiques, Université de Lille I, 59655, Villeneuve d’Ascq Cedex, France

    Mostafa Mbekhta

Authors
  1. Manuel González
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mostafa Mbekhta
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Manuel González.

Additional information

Communicated by L. Molnár

This work was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01). The first author was supported in part by MTM2016-76958, Spain.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

González, M., Mbekhta, M. Linear maps that preserve semi-Fredholm operators acting on Banach spaces. ActaSci.Math. 84, 137–149 (2018). https://doi.org/10.14232/actasm-017-327-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.14232/actasm-017-327-x

AMS Subject Classification (2010)

Key words and phrases

Search

Navigation