Abstract
A Hilbert space operator T is said to have property (ω 1) if σ a (T)\σ aw (T) ⊆ π00(T), where σ a (T) and σ aw (T) denote the approximate point spectrum and the Weyl essential approximate point spectrum of T respectively, and π 00(T) = {λ ∈ iso σ(T), 0 < dimN(T − λI) < ∞}. If σ a (T)\σ aw (T) = π00(T), we say T satisfies property (ω). In this note, we investigate the stability of the property (ω 1) and the property (ω) under compact perturbations, and we characterize those operators for which the property (ω 1) and the property (ω) are stable under compact perturbations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aiena, P., Peña, P.: Variation on Weyl theorem. J. Math. Anal. Appl. 324, 566–579 (2006)
Aiena, P.: Property (ω) and perturbation II. J. Math. Anal. Appl. 342, 830–837 (2008)
Aiena, P., Biondi, M. T.: Property (ω) and perturbations. J. Math. Anal. Appl. 336, 683–692 (2007)
Aiena, P. Biondi, M. T., Villafañe, F.: Property (ω) and perturbations III. J. Math. Anal. Appl. 353, 205–214 (2009)
Aiena, P.: Fredholm and Local Spectral Theory, with Applications to Multipliers, Kluwer Academic Publishers, Dordrecht, 2004
Cao, X. H., Guo, M. Z., Meng, B.: Weyl spectra and Weyl’s theorem. J. Math. Anal. Appl. 288, 758–767 (2003)
Harte, R., Lee, W. Y.: Another note on Weyl’s theorem. Trans. Amer. Math. Soc. 349, 2115–2124 (1997)
Herrero, D. A., Taylor, T. J., Wang, Z. Y.: Variation of the point spectrum under compact perturbations. Oper. Theory Adv. Appl. 32, 113–158 (1988)
Herrero, D. A.: Economical compact perturbations, II, Filling in the holes. J. Operator Theory, 19, 25–42 (1988)
Herrero, D. A.: Approximation of Hilbert Space Operators, Longman Scientific and Technical, Harlow, 1989
Ji, Y. Q.: Quasitriangular+small compact=strongly irreducible. Trans. Amer. Math. Soc. 351, 4657–4673 (1999)
Laursen, K. B., Neumann, M. M.: An Introduction to Local Spectral Theorey, Oxford University Press, New York, 2000
Oberai, K. K.: On the Weyl spectrum, II. Illinois J. Math. 21, 84–90 (1977)
Rakočeviĉ, V.: Operators obeying a-Weyl’s theorem. Rev. Roumaine Math. Pures Appl. 34, 915–919 (1989)
Rakočeviĉ, V.: On a class of operators. Mat. Vesnik, 37, 423–426 (1985)
Weyl, H.: Über beschränkte quadratische Formen, deren Differenz vollstetig ist. Rend. Circ. Mat. Palermo, 27, 373–392 (1909)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by the Fundamental Research Funds for the Central Universities (Grant No. GK201301007) and National Natural Science Foundation of China (Grant No. 11371012)
Rights and permissions
About this article
Cite this article
Shi, W.J., Cao, X.H. Property (ω) and its perturbations. Acta. Math. Sin.-English Ser. 30, 797–804 (2014). https://doi.org/10.1007/s10114-014-3023-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-014-3023-9