Continuity of approximate point spectrum on the algebra $B(X)$
Commun. Korean Math. Soc. 2013 Vol. 28, No. 3, 487-500
https://doi.org/10.4134/CKMS.2013.28.3.487
Printed July 1, 2013
Salvador S\'anchez-Perales and Victor A. Cruz-Barriguete
Instituto de F\'isica y Matem\'aticas, Instituto de F\'isica y Matem\'aticas
Abstract : In this paper we provide a brief introduction to the continuity of approximate point spectrum on the algebra $B(X)$, using basic properties of Fredholm operators and the SVEP condition. Also, we give an example showing that in general it not holds that if the spectrum is continuous an operator $T$, then for each $\lambda\in\sigma_{s-F}(T)\setminus\overline{\rho^{\pm}_{s-F}(T)}$ and $\epsilon>0$, the ball $B(\lambda,\epsilon)$ contains a component of $\sigma_{s-F}(T)$, contrary to what has been announced in [J. B. Conway and B. B. Morrel, Operators that are points of spectral continuity II, Integral Equations Operator Theory 4 (1981), 459--503] page 462.
Keywords : approximate point spectrum, continuity of the spectrum
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