Left and right generalized Drazin invertible operators and local spectral theory
DOI:
https://doi.org/10.22199/issn.0717-6279-2019-05-0058Keywords:
Left and right generalized Drazin invertible operators, Generalized Drazin invertible operators, SVEP, Local spectral theoryAbstract
In this paper, we give some characterizations of the left and right generalized Drazin invertible bounded operators in Banach spaces by means of the single-valued extension property (SVEP). In particular, we show that a bounded operator is left (resp. right) generalized Drazin invertible if and only if admits a generalized Kato decomposition and has the SVEP at 0 (resp. it admits a generalized Kato decomposition and its adjoint has the SVEP at 0. In addition, we prove that both of the left and the right generalized Drazin operators are invariant under additive commuting finite rank perturbations. Furthermore, we investigate the transmission of some local spectral properties from a bounded linear operator, as the SVEP, Dunford property (C), and property (β), to its generalized Drazin inverseReferences
P. Aiena, Fredholm and local spectral theory, with applications to multipliers. Dordrecht: Springer, 2004, doi: 10.1007/1-4020-2525-4.
P. Aiena and S. Triolo, “Local spectral theory for Drazin invertible operators”, Journal of mathematical analysis and applications, vol. 435, no. 1, pp. 414–424, Mar. 2016, doi: 10.1016/j.jmaa.2015.10.042.
S. Campbell, Singular systems of differential equations, vol. 1. London: Pitman, 1980.
S. Campbell, Singular systems of differential equations, vol. 2. London: Pitman, 1982.
G. Corach, B. Duggal, and R. Harte, “Extensions of Jacobsons Lemma”, Communications in algebra, vol. 41, no. 2, pp. 520–531, Feb. 2013, doi: 10.1080/00927872.2011.602274.
M. Drazin, “Pseudo-inverses in associative rings and semigroups”, The American mathematical monthly, vol. 65, no. 7, pp. 506–514, Sep. 1958, doi: 10.1080/00029890.1958.11991949.
Q. Jiang , H. Zhong, “Generalized Kato decomposition, single-valued extension property and approximate point spectrum”, Journal of mathematical analysis and applications, vol. 356, no. 1, pp. 322-327, Aug. 2009, doi: 10.1016/j.jmaa.2009.03.017.
M. Cvetković, “On upper and lower generalized Drazin invertible operators”, Functional analysis, approximation and computation, vol. 7, no. 3, pp. 67-74, 2015. [On line]. Available: https://bit.ly/2PoFRoe
M. González, M. Mbekhta and M. Oudghiri, “On the isolated points of the surjective spectrum of bounded operator”, Proceedings of the American mathematical society, vol. 136, no. 10, pp. 3521-3528, May 2008, doi: 10.1090/S0002-9939-08-09549-X.
R. Harte, “Spectral projections”, Irish mathematical society newsletter, vol. 11, pp. 10-15, Sep. 1984. [On line]. Available: https://bit.ly/36z8ll1
M. Kaashoek and D. Lay, “Ascent, descent, and commuting perturbations”, Transactions of the American mathematical society, vol. 169, pp. 35–47, 1972, doi: 10.1090/S0002-9947-1972-0312299-8.
T. Kato, Perturbation theory for linear operators, New York: Springer, 1966.
N. Khaldi, M. Benharrat, B. Messirdi, “Linear boundary value problems described by Drazin invertible operators”, Mathematical notes, vol. 101, no. 5-6, pp. 994-999, Jun. 2017, doi: 10.1134/S0001434617050261.
J. Koliha, “A generalized Drazin inverse”, Glasgow mathematical journal, vol. 38, no. 3, pp. 367-381, Sep. 1996, doi: 10.1017/S0017089500031803.
J. Koliha and T. Tran, “Semistable operators and singularly perturbed differential equations”, Journal of mathematical analysis and applications, vol. 231, no. 2, pp. 446-458, Mar. 1999, doi: 10.1006/jmaa.1998.6235.
J. Koliha and T. Tran, “The Drazin inverse for closed linear operators and the asymptotic convergence of C0-semigroups”, Journal of operator theory, vol. 46, no. 2, pp. 323-336, 2001. [On line]. Available: https://bit.ly/2RVZx4u
J. Labrousse, “Les opérateurs quasi-Fredholm une généralisation des opérateurs semi-Fredholm”, Rendiconti del circolo matematico di Palermo, vol. 2, no. 29, pp. 161-258, May 1980, doi: 10.1007/BF02849344
M. Lahrouz, M. Zohry “Weyl type theorems and the approximate point spectrum”, Bulletin - Irish mathematical society, no, 55, pp. 41-51, (2005). [On line]. Available: https://bit.ly/2LYJqzg
K. Laursen and M. Neumann, An introduction to local spectral theory. Oxford: Clarendon Press, 2000.
K. Hocine, M. Benharrat and B. Messirdi, “Left and right generalized Drazin invertible operators”, Linear and multilinear algebra, vol. 63, no. 8, pp. 1635-1648, Sep. 2015, doi: 10.1080/03081087.2014.962534.
P. Vrbová, “On local spectral properties of operators in Banach spaces”, Czechoslovak Mathematical Journal, vol. 23, no. 3, pp. 483-492, 1973. [On line]. Available: https://bit.ly/2RXwVI7
Published
How to Cite
Issue
Section
-
Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- No additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.
