# THE SPECTRAL CONTINUITY OF ESSENTIALLY HYPONORMAL OPERATORS

• Kim, An-Hyun (Department of Mathematics Changwon National University) ;
• Ryu, Eun-Jin (Department of Mathematics Changwon National University)
• Received : 2013.11.03
• Published : 2014.07.31
• 69 4

#### Abstract

If A is a unital Banach algebra, then the spectrum can be viewed as a function ${\sigma}$ : 𝕬 ${\rightarrow}$ 𝕾, mapping each T ${\in}$ 𝕬 to its spectrum ${\sigma}(T)$, where 𝕾 is the set, equipped with the Hausdorff metric, of all compact subsets of $\mathbb{C}$. This paper is concerned with the continuity of the spectrum ${\sigma}$ via Browder's theorem. It is shown that ${\sigma}$ is continuous when ${\sigma}$ is restricted to the set of essentially hyponormal operators for which Browder's theorem holds, that is, the Weyl spectrum and the Browder spectrum coincide.

#### Keywords

spectrum;essential spectrum;spectral continuity;Weyl's theorem;Browder's theorem

#### Acknowledgement

Supported by : Changwon National University

#### References

1. S. K. Berberian, An extension of Weyl's theorem to a class of not necessarily normal operators, Michigan Math. J. 16 (1969), 273-279. https://doi.org/10.1307/mmj/1029000272
2. R. H. Bouldin, The essential minimum modulus, Indiana Univ. Math. J. 30 (1981), no. 4, 513-517. https://doi.org/10.1512/iumj.1981.30.30042
3. R. H. Bouldin, Distance to invertible linear operators without separability, Proc. Amer. Math. Soc. 116 (1992), no. 2, 489-497. https://doi.org/10.1090/S0002-9939-1992-1097336-6
4. A. Bottcher, S. Grudsky, and I. Spitkovsky, The spectrum is discontinuous on the manifold of Toeplitz operators, Arch. Math. 75 (2000), no. 1, 46-52. https://doi.org/10.1007/s000130050472
5. R. G. Douglas, Banach Algebra Techniques in Operator Theory, Academic press, New York, 1972.
6. A. Bottcher and B. Silbermann, Analysis of Toeplitz Operators, Springer, Berlin-Heidelberg-New York, 1990.
7. L. A. Coburn, Weyl's theorem for nonnormal operators, Michigan Math. J. 13 (1966), 285-288. https://doi.org/10.1307/mmj/1031732778
8. 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
9. D. R. Farenick and W. Y. Lee, Hyponormality and spectra of Toeplitz operators, Trans. Amer. Math. Soc. 348 (1996), no. 10, 4153-4174. https://doi.org/10.1090/S0002-9947-96-01683-2
10. P. R. Halmos, A Hilbert Space Problem Book, Springer, New York, 1982.
11. R. E. Harte, Invertibility and Singularity for Bounded Linear Operators, Dekker, New York, 1988.
12. 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
13. D. A. Herrero, Economical compact perturbations I: Erasing normal eigenvalues, J. Operator Theory 10 (1983), no. 2, 289-306.
14. I. S. Hwang and W. Y. Lee, On the continuity of spectra of Toeplitz operators, Arch. Math. 70 (1998), no. 1, 66-73. https://doi.org/10.1007/s000130050166
15. I. S. Hwang and W. Y. Lee, The spectrum is continuous on the set of p-hyponormal operators, Math. Z. 235 (2000), no. 1, 151-157. https://doi.org/10.1007/s002090000128
16. J. D. Newburgh, The variation of spectra, Duke Math. J. 18 (1951), 165-176. https://doi.org/10.1215/S0012-7094-51-01813-3
17. N. K. Nikolskii, Treatise on the Shift Operator, Springer, New York, 1986.
18. C. M. Pearcy, Some Recent Developments in Operator Theory, CBMS 36, Providence: AMS, 1978.
19. H. Weyl, Uber beschrankte quadratische Formen, deren Differenz vollsteig ist, Rend. Circ. Mat. Palermo 27(1909), 373-392. https://doi.org/10.1007/BF03019655
20. H. Widom, On the spectrum of a Toeplitz operator, Pacific J. Math. 14 (1964), 365-375. https://doi.org/10.2140/pjm.1964.14.365
21. A. H. Kim and E. Y. Kwon, Spectral continuity of essentially p-hyponormal operators, Bull. Korean Math. Soc. 43 (2006), no. 2, 389-393. https://doi.org/10.4134/BKMS.2006.43.2.389
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9419931173324585, "perplexity": 1425.2418651163673}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540534443.68/warc/CC-MAIN-20191212000437-20191212024437-00545.warc.gz"}