Korean J. Math. Vol. 23 No. 4 (2015) pp.601-605
DOI: https://doi.org/10.11568/kjm.2015.23.4.601

A note on spectral continuity

Main Article Content

In Ho Jeon
In Hyoun Kim

Abstract

In the present note, provided $T\in\mathscr{L(H)}$ is biquasitriangular and Browder's theorem hold for $T$, we show that the spectrum $\sigma$ is continuous at $T$ if and only if the essential spectrum $\sigma_{e}$ is continuous at $T$.


Article Details

Supporting Agencies

This work was supported by the Incheon National University Research Grant in 2013

References

[1] C. Apostol, C. Foias and D. Voiculescu, Some results on non-quasitriangular operators. IV, Rev. Roum. Math. Pures Appl. 18 (1973), 487–514. Google Scholar

[2] J. B. Conway and B. B. Morrel, Operators that are points of spectral continuity, Integral equations and operator theory, 2 (1979), 174-198. Google Scholar

[3] J. B. Conway and B. B. Morrel, Operators that are points of spectral continuity II, Integral equations and operator theory 4 (1981), 459-503. Google Scholar

[4] N. Dunford, Spectral operators, Pacific J. Math. 4 (1954), 321–354. Google Scholar

[5] S. V. Djordjevi c and B. P. Duggal, Weyl's theorem and continuity of spectra in the class of p-hyponormal operators, Studia Math. 143 (2000), 23-32. Google Scholar

[6] S. V.Djordjevi c and Y. M. Han, Browder's theorem and spectral continuity, Glasgow Math. J. 42 (2000), 479-486. Google Scholar

[7] S. V.Djordjevi c, Continuity of the essential spectrum in the class of quasihyponormal operators, Vesnik Math. 50 (1998) 71-74. Google Scholar

[8] B. P. Duggal, I. H. Jeon, and I. H. Kim, Continuity of the spectrum on a class of upper triangular operator matrices, Jour. Math. Anal. Appl. 370 (2010), 584–587. Google Scholar

[9] R. Harte and W. Y. Lee, Another note on Weyl’s theorem, Trans. Amer. Math. Soc. 349 (1997), 2115–2124. Google Scholar

[10] I. S. Hwang and W. Y. Lee, The spectrum is continuous on the set of phyponormal operators, Math. Z. 235 (2000), 151–157. Google Scholar

[11] R. Lange, Biquasitriangularity and spectral continuity, Glasgow Math. J. 26 (1985), 177–180. Google Scholar

[12] J. D. Newburgh, The variation of spectra, Duke Math. J. 18 (1951), 165–176. Google Scholar