CONTINUITY OF APPROXIMATE POINT SPECTRUM ON THE ALGEBRA B(X)

• Published : 2013.07.31
• 21 4

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 ${\in}$ > 0, the ball $B({\lambda},{\in})$ 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

References

1. M. Ahues, A. Largillier, and B. V. Limaye, Spectral Computations for Bounded Operators, Chapman & Hall/CRC, 2001.
2. P. Aiena, Fredholm and Local Spectral Theory with Applications to Multipliers, Kluwer Acad., 2004.
3. C. Apostol, L. A. Fialkow, D. A. Herrero, and D. Voiculescu, Approximation of Hilbert space operators. Vol. II, Res. Notes Math. 102, Pitman, Boston, 1984.
4. L. Burlando, Continuity of spectrum and spectral radius in algebras of operators, Ann. Fac. Sci. Toulouse Math. 9 (1988), no. 1, 5-54. https://doi.org/10.5802/afst.647
5. S. R. Caradus, W. E. Pfaffenberger, and B. Yood, Calkin Algebras and Algebras of Operators on Banach Spaces, Marcel Dekker, 1974.
6. J. B. Conway and B. B. Morrel, Operators that are points of spectral continuity, Integral Equations Operator Theory 2 (1979), no. 2, 174-198. https://doi.org/10.1007/BF01682733
7. J. B. Conway and B. B. Morrel, Operators that are points of spectral continuity. II, Integral Equations Operator Theory 4 (1981), no. 4, 459-503. https://doi.org/10.1007/BF01686497
8. S. V. Djordjevic, The continuity of the essential approximative point spectrum, Facta Univ. Ser. Math. Inform. 10 (1995), 97-104.
9. S. V. Djordjevic and Y. M. Han, Browder's theorems and spectral continuity, Glasg. Math. J. 42 (2000), no. 3, 479-486. https://doi.org/10.1017/S0017089500030147
10. S. V. Djordjevic and Y. M. Han, Operator matrices and spectral continuity, Glasg. Math. J. 43 (2001), no. 3, 487-490.
11. B. P. Duggal, I. H. Jeon, and I. H. Kim, Continuity of the spectrum on a class of upper triangular operator matrices, J. Math. Anal. Appl. 370 (2010), 584-587. https://doi.org/10.1016/j.jmaa.2010.04.068
12. P. R. Halmos and G. Lumer, Square roots of operators. II, Proc. Amer. Math. Soc. 5 (1954), 589-595. https://doi.org/10.1090/S0002-9939-1954-0062953-5
13. R. E. Harte and W. Y. Lee, Another note on Weyl's theorem, Trans. Amer. Math. Soc. 349 (1997), no. 5, 2115-2124. https://doi.org/10.1090/S0002-9947-97-01881-3
14. J. D. Newburgh, The variation of Spectra, Duke Math. J. 18 (1951), 165-176. https://doi.org/10.1215/S0012-7094-51-01813-3
15. S. Sanchez-Perales and S. V. Djordjevic, Continuity of spectrum and approximate point spectrum on operator matrices, J. Math. Anal. Appl. 378 (2011), no. 1, 289-294. https://doi.org/10.1016/j.jmaa.2011.01.062

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