Browse > Article
http://dx.doi.org/10.4134/BKMS.2012.49.5.899

GENERALIZED WEYL'S THEOREM FOR FUNCTIONS OF OPERATORS AND COMPACT PERTURBATIONS  

Zhou, Ting Ting (Institute of Mathematics Jilin University)
Li, Chun Guang (Institute of Mathematics Jilin University)
Zhu, Sen (Department of Mathematics Jilin University, School of Mathematical Sciences Fudan University)
Publication Information
Bulletin of the Korean Mathematical Society / v.49, no.5, 2012 , pp. 899-910 More about this Journal
Abstract
Let $\mathcal{H}$ be a complex separable infinite dimensional Hilbert space. In this paper, a necessary and sufficient condition is given for an operator T on $\mathcal{H}$ to satisfy that $f(T)$ obeys generalized Weyl's theorem for each function $f$ analytic on some neighborhood of ${\sigma}(T)$. Also we investigate the stability of generalized Weyl's theorem under (small) compact perturbations.
Keywords
generalized Weyl's theorem; operator approximation; compact perturbations;
Citations & Related Records
연도 인용수 순위
  • Reference
1 M. Berkani and M. Sarih, On semi B-Fredholm operators, Glasg. Math. J. 43 (2001), no. 3, 457-465.
2 X. H. Cao, Topological uniform descent and Weyl type theorem, Linear Algebra Appl. 420 (2007), no. 1, 175-182.   DOI   ScienceOn
3 X. H. Cao, M. Z. Guo, and B. Meng, Weyl type theorems for p-hyponormal and M- hyponormal operators, Studia Math. 163 (2004), no. 2, 177-188.   DOI
4 L. A. Coburn, Weyl's theorem for nonnormal operators, Michigan Math. J. 13 (1966), 285-288.   DOI
5 J. B. Conway, A course in Functional Analysis, second ed., Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1990.
6 R. E. Curto and Y. M. Han, Generalized Browder's and Weyl's theorems for Banach space operators, J. Math. Anal. Appl. 336 (2007), no. 2, 1424-1442.   DOI   ScienceOn
7 B. P. Duggal, Hereditarily polaroid operators, SVEP and Weyl's theorem, J. Math. Anal. Appl. 340 (2008), no. 1, 366-373.   DOI   ScienceOn
8 N. Dunford and J. T. Schwartz, Linear Operators. Part I, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1988.
9 D. A. Herrero, Economical compact perturbations. II. Filling in the holes, J. Operator Theory 19 (1988), no. 1, 25-42.
10 D. A. Herrero, Approximation of Hilbert Space Operators. Vol. 1, second ed., Pitman Research Notes in Mathematics Series, vol. 224, Longman Scientific & Technical, Harlow, 1989.
11 M. Berkani, On the equivalence of Weyl theorem and generalized Weyl theorem, Acta Math. Sin. (Engl. Ser.) 23 (2007), no. 1, 103-110.   DOI
12 M. Berkani and A. Arroud, Generalized Weyl's theorem and hyponormal operators, J. Aust. Math. Soc. 76 (2004), no. 2, 291-302.   DOI
13 M. Berkani and J. J. Koliha, Weyl type theorems for bounded linear operators, Acta Sci. Math. (Szeged) 69 (2003), no. 1-2, 359-376.
14 H. Weyl, Uber beschrankte quadratische formen, deren differenz, vollsteig ist, Rend. Circ. Mat. Palermo 27 (1909), 373-392.   DOI
15 H. Zguitti, A note on generalized Weyl's theorem, J. Math. Anal. Appl. 316 (2006), no. 1, 373-381.   DOI   ScienceOn
16 S. Zhu and C. G. Li, SVEP and compact perturbations, J. Math. Anal. Appl. 380 (2011), no. 1, 69-75.   DOI   ScienceOn
17 C. G. Li, S. Zhu, and Y. L. Feng, Weyl's theorem for functions of operators and ap- proximation, Integral Equations Operator Theory 67 (2010), no. 4, 481-497.   DOI
18 H. Radjavi and P. Rosenthal, Invariant Subspaces, second ed., Dover Publications Inc., Mineola, NY, 2003.
19 P. Aiena and M. T. Biondi, Property (w) and perturbations, J. Math. Anal. Appl. 336 (2007), no. 1, 683-692.   DOI   ScienceOn
20 P. Aiena, M. T. Biondi, and F. Villafane, Property (w) and perturbations. III, J. Math. Anal. Appl. 353 (2009), no. 1, 205-214.   DOI   ScienceOn
21 M. Amouch, Weyl type theorems for operators satisfying the single-valued extension property, J. Math. Anal. Appl. 326 (2007), no. 2, 1476-1484.   DOI   ScienceOn
22 I. J. An and Y. M. Han, Weyl's theorem for algebraically quasi-class A operators, Integral Equations Operator Theory 62 (2008), no. 1, 1-10.   DOI
23 S. K. Berberian, An extension of Weyl's theorem to a class of not necessarily normal operators, Michigan Math. J. 16 (1969), 273-279.   DOI
24 M. Berkani, Index of B-Fredholm operators and generalization of a Weyl theorem, Proc. Amer. Math. Soc. 130 (2002), no. 6, 1717-1723 (electronic).   DOI   ScienceOn
25 P. Aiena and M. Berkani, Generalized Weyl's theorem and quasiaffinity, Studia Math. 198 (2010), no. 2, 105-120.   DOI
26 P. Aiena, Property (w) and perturbations. II, J. Math. Anal. Appl. 342 (2008), no. 2, 830-837.   DOI   ScienceOn