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

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Abstract

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.

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References

  1. Aiena P.: Fredholm and Local Spectral Theory, with Applications to Multipliers. Kluwer Publisher, Dordrecht (2004)

    MATH  Google Scholar 

  2. Aupetit B.: A Primer on Spectral Theory. Springer-Verlag, New York (1991)

    MATH  Google Scholar 

  3. Bourhim A., Miller V.G.: Linear maps on \({\mathcal{M}_{n} }\) preserving the local spectral radius. Studia Math. 188, 67–75 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bourhim A., Ransford T.J.: Additive maps preserving local spectrum. Integr. Equ. Oper. Theory 55, 377–385 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bračič J., Müller V.: Local spectrum and local spectral radius of an operator at a fixed vector. Studia Math. 194, 155–162 (2009)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  7. González M., Mbekhta M.: Linear maps on \({\mathcal{M}_{n} }\) preserving the local spectrum. Linear Algebra Appl. 427, 176–182 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  8. V. Müller, Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras, 2nd ed., Birkhäuser, 2007.

  9. T. Ransford, Potential Theory in the Complex Plane, Cambridge University Press, 1995.

  10. Vrbová P.: On local spectral properties of operators in Banach spaces. Czechos lovak Math. J. 23, 483–492 (1973)

    Google Scholar 

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Authors and Affiliations

  1. Faculty of Mathematics and Informatics, Ovidius University, Mamaia Blvd. 124, Constanţa, Romania

    Constantin Costara

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  1. Constantin Costara
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Correspondence to Constantin Costara.

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This work was supported by CNCSIS-UEFISCSU, project number 24/11.08.2010, PN II-RU Code 300/2010.

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Costara, C. Automatic continuity for linear surjective mappings decreasing the local spectral radius at some fixed vector. Arch. Math. 95, 567–573 (2010). https://doi.org/10.1007/s00013-010-0191-4

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  • DOI: https://doi.org/10.1007/s00013-010-0191-4

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