Archiv der Mathematik

, Volume 95, Issue 6, pp 567–573 | Cite as

Automatic continuity for linear surjective mappings decreasing the local spectral radius at some fixed vector



Let X be a complex Banach space and denote by \({\mathcal{L}\left( X\right)}\) the space of bounded linear operators on X. Let e be a nonzero element of X. We prove that if φ is a linear and surjective mapping from \({ \mathcal{L}\left( X\right) }\) into itself which decreases the local spectral radius at e, then φ is automatically continuous.

Mathematics Subject Classification (2010)

Primary 46H40 Secondary 47A11 


Automatic continuity Local spectrum Local spectral radius 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aiena P.: Fredholm and Local Spectral Theory, with Applications to Multipliers. Kluwer Publisher, Dordrecht (2004)MATHGoogle Scholar
  2. 2.
    Aupetit B.: A Primer on Spectral Theory. Springer-Verlag, New York (1991)MATHGoogle Scholar
  3. 3.
    Bourhim A., Miller V.G.: Linear maps on \({\mathcal{M}_{n} }\) preserving the local spectral radius. Studia Math. 188, 67–75 (2008)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bourhim A., Ransford T.J.: Additive maps preserving local spectrum. Integr. Equ. Oper. Theory 55, 377–385 (2006)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bračič J., Müller V.: Local spectrum and local spectral radius of an operator at a fixed vector. Studia Math. 194, 155–162 (2009)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Brešar M., Šemrl P.: Linear maps preserving the spectral radius. J. Funct. Anal. 142, 360–368 (1996)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    González M., Mbekhta M.: Linear maps on \({\mathcal{M}_{n} }\) preserving the local spectrum. Linear Algebra Appl. 427, 176–182 (2007)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    V. Müller, Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras, 2nd ed., Birkhäuser, 2007.Google Scholar
  9. 9.
    T. Ransford, Potential Theory in the Complex Plane, Cambridge University Press, 1995.Google Scholar
  10. 10.
    Vrbová P.: On local spectral properties of operators in Banach spaces. Czechos lovak Math. J. 23, 483–492 (1973)Google Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  1. 1.Faculty of Mathematics and InformaticsOvidius UniversityConstanţaRomania

Personalised recommendations