On the perturbation functions and similarity orbits
Volume 188 / 2008
Studia Mathematica 188 (2008), 57-66
MSC: Primary 47A53; Secondary 47A30.
DOI: 10.4064/sm188-1-3
Abstract
We show that the essential spectral radius $ \varrho_e (T)$ of $T\in B(H)$ can be calculated by the formula $ \varrho_e (T) = \inf \{ {\cal{F}}_{\sharp \cdot\sharp } (X T X^{-1}) :X$ an invertible operator$\},$ where ${\cal{F}}_{\sharp \cdot\sharp } (T)$ is a ${\mit\Phi}_1$-perturbation function introduced by Mbekhta [J. Operator Theory 51 (2004)]. Also, we show that if ${\cal{G}}_{\sharp \cdot\sharp } (T)$ is a ${\mit\Phi}_2$-perturbation function [loc. cit.] and if $T$ is a Fredholm operator, then $ \mathop{\rm dist}(0,\sigma_e(T)) = \sup \{ {\cal{G}}_{\sharp \cdot\sharp } (X T X^{-1}): X$ an invertible operator$\}.$
