Local spectrum and local spectral radius of an operator at a fixed vector
Tom 194 / 2009
Studia Mathematica 194 (2009), 155-162
MSC: Primary 47A11; Secondary 47B49.
DOI: 10.4064/sm194-2-3
Streszczenie
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 $e$.
