# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

## 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 $e$.

#### Autorzy

• Janko BračičIMFM, University of Ljubljana
SI-1000 Ljubljana, Slovenia
e-mail