Local spectrum and local spectral radius of an operator at a fixed vector

Volume 194 / 2009

Janko Bračič, Vladimír Müller Studia Mathematica 194 (2009), 155-162 MSC: Primary 47A11; Secondary 47B49. DOI: 10.4064/sm194-2-3


Let $\mathscr X$ be a complex Banach space and $e\in\mathscr X$ a nonzero vector. Then the set of all operators $T\in{\cal L}(\mathscr X)$ with $\sigma_T(e)=\sigma_\delta(T)$, respectively $r_T(e)=r(T)$, is residual. This is an analogy to the well known result for a fixed operator and variable vector. The results are then used to characterize linear mappings preserving the local spectrum (or local spectral radius) at a fixed vector $e$.


  • Janko BračičIMFM, University of Ljubljana
    Jadranska ul. 19
    SI-1000 Ljubljana, Slovenia
  • Vladimír MüllerMathematical Institute
    Czech Academy of Sciences
    Žitná 25
    115 67 Praha 1, Czech Republic

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