Stability of Property (gm) under Perturbation and Spectral Properties Type Weyl Theorems

Authors: M. H. M. Rashid

Abstract:

A Banach space operator T obeys property (gm) if the isolated points of the spectrum σ(T) of T which are eigenvalues are exactly those points λ of the spectrum for which T − λI is a left Drazin invertible. In this article, we study the stability of property (gm), for a bounded operator acting on a Banach space, under perturbation by finite rank operators, by nilpotent operators, by quasi-nilpotent operators, or more generally by algebraic operators commuting with T.

Keywords: Weyl’s theorem, Weyl spectrum, polaroid operators, property (gm), property (m).

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.2571940

References:

[1] P. Aiena, Fredholm and local spectral theory with applications to multipliers,Kluwer Acad. Publishers, Dordrecht, 2004.
[2] P. Aiena, Quasi Fredholm operators and Localized SVEP, Acta Sci. Math. (Szeged) 73(2007), 251–263.
[3] P. Aiena, T. L. Miller, On generalized a-Browders theorem, Stud. Math. 180 (3) (2007), 285–300.
[4] P. Aiena, E. Aponte and E. Bazan, Weyl type theorems for left and right polaroid operators, Integral Equations Operator Theory 66 (2010), 1–20.
[5] P. Aiena , J. R. Guillen and P. Pe˜na, Property (w) for perturbations of polaroid operators, Linear Alg. Appl. 428 (2008), 1791-1802.
[6] P. Aiena , J. R. Guill´en and P. Pe˜na, Property (gR) and perturbations, Acta Sci. Math. (Szeged) 78(2012), 569–588.
[7] P. Aiena, E. Aponte, J. R. Guill´en and P. Pe˜na, Property (R) under perturbations, Mediterr. J. Math. 10 (1) (2013), 367–382.
[8] P. Aiena, E. Aponte, Polaroid type operators under perturbations, Studia Math. 214, (2), (2013), 121-136.
[9] M. Amouch, H. Zguitti, On the equivalence of Browder’s and generalized Browder’s theorem, Glasg. Math. J. 48 (2006), 179185.
[10] M. Amouch, M. Berkani, on the property (gw), Mediterr. J. Math. 5 (2008), 371-378.
[11] M. Berkani, On a class of quasi-Fredholm operators, Integral Equations and Operator Theory. 34 (1999), no.2, 244–249.
[12] M. Berkani, M. Sarih, On semi B-Fredholm operators, Glasg. Math. J. 43 (2001), 457–465.
[13] M. Berkani, Index of B-Fredholm operators and gereralization of a Weyl Theorem, Proc. Amer. Math. Soc. 130 (2001), 1717-1723.
[14] M. Berkani, B-Weyl spectrum and poles of the resolvent, J. Math. Anal. Appl. 272 (2002), 596–603.
[15] M. Berkani and J. Koliha, Weyl type theorems for bounded linear operators, Acta Sci. Math. 69 (2003), 359–376.
[16] M. Berkani, On the equivalence of Weyl theorem and generalized Weyl theorem, Acta Math. Sinica 272 (2007), 103–110.
[17] M. Berkani and H. Zariouh, Generalized a-Weyl’s theorem and perturbations, Functional Analysis, Approximation and computation 2 (1)(2010), 7–18.
[18] M. Berkani and H. Zariouh, Perturbation results for Weyl type theorem, Acta Math. Univ. Comenianae 80(2011), 119–132.
[19] L. A. Coburn, Weyl’s theorem for nonnormal operators, Michigan Math. J. 13 (1966), 285–288.
[20] J. K. Finch, The single valued extension property on a Banach space, Pacific J. Math. 58 (1975), 61–69.
[21] H. Heuser, Functional analysis, Marcel Dekker, New York, 1982.
[22] K. B. Laursen, Operators with finite ascent, Pacific J. Math. 152 (1992), 323–336.
[23] K. B. Laursen, M. M. Neumann, An introduction to local spectral theory, Oxford. Clarendon, 2000.
[24] W. Y. Lee, S. H. Lee, On Weyls theorem II, Math. Japonica 43 (1996), 549-553.
[25] V. Rakoˇcevi´c, Operators obeying a-Weyl’s theorem, Rev. Roumaine Math. Pures Appl. 10(1986), 915–919.
[26] V. Rako˜cevi´c, Semi-Browder operators and perturbations, Studia Math. 122(1997), 131–137.
[27] M. H. M. Rashid, Properties (t) and (gt) For Bounded Linear Operators , Mediterr. J. Math. (2014) 11: 729. doi:10.1007/s00009-013-0313-x.
[28] M. H. M. Rashid, Properties (m) and (gm) For Bounded Linear Operators , Jordan Journal of Mathematics and Statistics 6(2)(2013), 81–102.
[29] H. O. Tylli, On the asymptotic behaviour of some quantities related to semi-Fredholm operators, J. London Math. Soc. 31(1985), 340–348.
[30] Q. Zeng, Q. Jiang, and H. Zhong, Spectra originated from semi-B-Fredholm theory and commuting perturbations, arXiv:1203.2442vl
[math. FA] 12 Mar 2012.