Hostname: page-component-77c89778f8-gvh9x Total loading time: 0 Render date: 2024-07-21T17:02:29.254Z Has data issue: false hasContentIssue false
  • English
  • Français

WEYL TYPE THEOREMS FOR FUNCTIONS OF OPERATORS

Published online by Cambridge University Press:  30 March 2012

SEN ZHU ,
CHUN GUANG LI  and
TING TING ZHOU
SEN ZHU
Affiliation:
Department of Mathematics, Jilin University, Changchun 130012, P.R. China e-mail: [email protected]
CHUN GUANG LI
Affiliation:
Institute of Mathematics, Jilin University, Changchun 130012, P.R. China e-mail: [email protected]
TING TING ZHOU
Affiliation:
Institute of Mathematics, Jilin University, Changchun 130012, P.R. China e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A-Weyl's theorem and property (ω), as two variations of Weyl's theorem, were introduced by Rakočević. In this paper, we study a-Weyl's theorem and property (ω) for functions of bounded linear operators. A necessary and sufficient condition is given for an operator T to satisfy that f(T) obeys a-Weyl's theorem (property (ω)) for all f ∈ Hol(σ(T)). Also we investigate the small-compact perturbations of operators satisfying a-Weyl's theorem (property (ω)) in the setting of separable Hilbert spaces.

Keywords

Primary 47A5347A60Secondary 47A10
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 54 , Issue 3 , September 2012 , pp. 493 - 505
Copyright
Copyright © Glasgow Mathematical Journal Trust 2012

References

REFERENCES

1.Aiena, P., Fredholm and local spectral theory, with applications to multipliers (Kluwer Academic Publishers, Dordrecht, 2004).Google Scholar
2.Aiena, P., Classes of operators satisfying a-Weyl's theorem, Studia Math. 169 (2) (2005), 105–122.CrossRefGoogle Scholar
3.Aiena, P., Property (w) and perturbations. II, J. Math. Anal. Appl. 342 (2) (2008), 830–837.CrossRefGoogle Scholar
4.Aiena, P., Biondi, M. T. and Villafañe, F., Property (w) and perturbations. III, J. Math. Anal. Appl. 353 (1) (2009), 205–214.CrossRefGoogle Scholar
5.Aiena, P. and Peña, P., Variations on Weyl's theorem, J. Math. Anal. Appl. 324 (1) (2006), 566–579.CrossRefGoogle Scholar
6.An, I. J. and Han, Y. M., Weyl's theorem for algebraically quasi-class A operators, Int. Equ. Oper. Theory 62 (1) (2008), 1–10.CrossRefGoogle Scholar
7.Barnes, B. A., Riesz points and Weyl's theorem, Int. Equ. Oper. Theory 34 (2) (1999), 187–196.CrossRefGoogle Scholar
8.Chō, M. and Han, Y. M., Riesz idempotent and algebraically M-hyponormal operators, Int. Equ. Oper. Theory 53 (3) (2005), 311–320.CrossRefGoogle Scholar
9.Coburn, L. A., Weyl's theorem for nonnormal operators, Michigan Math. J. 13 (1966), 285–288.CrossRefGoogle Scholar
10.Conway, J. B., A course in functional analysis, second ed. Graduate Texts in Mathematics, vol. 96, (Springer-Verlag, New York, 1990).Google Scholar
11.Curto, R. E. and Han, Y. M., Weyl's theorem, a-Weyl's theorem, and local spectral theory, J. London Math. Soc. (2) 67 (2) (2003), 499–509.CrossRefGoogle Scholar
12.Curto, R. E. and Han, Y. M., Weyl's theorem for algebraically paranormal operators, Int. Equ. Oper. Theory 47 (3) (2003), 307–314.CrossRefGoogle Scholar
13.Duggal, B. P., The Weyl spectrum of p-hyponormal operators, Int. Equ. Oper. Theory 29 (2) (1997)197–201.CrossRefGoogle Scholar
14.Han, Y. M. and Lee, W. Y., Weyl's theorem holds for algebraically hyponormal operators, Proc. Amer. Math. Soc. 128 (8) (2000), 2291–2296 (electronic).CrossRefGoogle Scholar
15.Herrero, D. A., Approximation of Hilbert space operators, vol. 1, second ed., Pitman Research Notes in Mathematics Series, vol. 224 (Longman Scientific & Technical, Harlow, 1989).Google Scholar
16.Herrero, D. A., Taylor, T. J. and Wang, Z. Y., Variation of the point spectrum under compact perturbations, in Topics in operator theory, Vol. 32 of Oper. Theory Adv. Appl., Birkhäuser, Basel, 1988, pp. 113–158.CrossRefGoogle Scholar
17.Li, C. G., Zhu, S. and Feng, Y. L., Weyl's theorem for functions of operators and approximation, Int. Equ. Oper. Theory 67 (4) (2010), 481–497.CrossRefGoogle Scholar
18.Oudghiri, M., Weyl's and Browder's theorems for operators satisfying the SVEP, Studia Math. 163 (1) (2004), 85–101.CrossRefGoogle Scholar
19.Radjavi, H. and Rosenthal, P., Invariant subspaces, second ed. (Dover Publications, Mineola, NY, 2003).Google Scholar
20.Rakočević, V., On one subset of M. Schechter's essential spectrum, Mat. Vesnik 5 (18)(33)(4) (1981), 389–391.Google Scholar
21.Rakočević, V., On the essential approximate point spectrum. II, Mat. Vesnik 36 (1) (1984), 89–97.Google Scholar
22.Rakočević, V., On a class of operators, Mat. Vesnik 37 (4) (1985), 423–426.Google Scholar
23.Rakočević, V., Approximate point spectrum and commuting compact perturbations, Glasgow Math. J. 28 (2) (1986), 193–198.CrossRefGoogle Scholar