Further common local spectral properties for bounded linear operators

Keywords: Jacobson's lemma, Common properties, Local spectral theory

Abstract

We study common local spectral properties for bounded linear operators A ∈ ℒ(X,Y) and B,C ∈ ℒ (Y,X) such that A(BA)2=ABACA=ACABA=(AC)2A. We prove that AC and BA share the single valued extension property, the Bishop property (β), the property (βε), the decomposition property (δ) and decomposability. Closedness of analytic core and quasinilpotent part are also investigated. Some applications to Fredholm operators are given.

Author Biography

Hassane Zguitti, Sidi Mohamed Ben Abdellah University.
Dept. of Mathematics.

References

P. Aiena, Fredholm and local spectral theory, with applications to multipliers. Dordrecht: Kluwer Academic Publishers, 2004, doi: 10.1007/1-4020-2525-4.

P. Aiena and M. González, “On the Dunford property (C) for bounded linear operators RS and SR”, Integral equations and operator theory, vol. 70, no. 4. pp. 561-568, Aug. 2011, doi: 10.1007/s00020-011-1875-2.

P. Aiena and M. Gonzalez, “Local spectral theory for operators R and S satisfying RSR=R2”, Extracta mathematicae, vol. 31, no. 1, pp. 37-46, 2016. [On line]. Available: https://bit.ly/2GRzox3

E. Albrecht and J. Eschemeier, “Analytic functional models and local spectral theory”, Proceedings of the London mathematical society, vol. 75, no. 2, pp. 323-348, Sep. 1997, doi: 10.1112/S0024611597000373.

B. A. Barnes, “Common operator properties of the linear operators RS and SR”, Proceedings of the american mathematical society, vol. 126, no. 4, pp. 1055-1061, Apr. 1998, doi: 10.1090/S0002-9939-98-04218-X.

C. Benhida and E. H. Zerouali, “Local spectral theory of linear operators RS and SR”. Integral equations and operator theory, vol. 54, no. 1, pp. 1-8, Jan. 2006, doi: 10.1007/s00020-005-1375-3.

H. Chen and M. Sheibani, “Cline's formula for g-Drazin inverses”, May 2018, Arxiv:1805.06133v1

G. Corach, B. Duggal and R. Harte, “Extensions of Jacobson's lemma”, Communications in algebra, vol. 41, no. 2, pp. 520-531, Jun. 2013, doi: 10.1080/00927872.2011.602274.

B. P. Duggal, “Operator equations ABA=A2 and BAB=B2”, Functional analysis, approximation and computation, vol. 3, no. 1, pp. 9-18, 2011. [On line]. Available: https://bit.ly/2vPggxw

J. Eschmeier and M. Putinar, “Bishop's condition (β) and rich extensions of linear operators”, Indiana University mathematics journal, vol. 37, no. 2, pp. 325-348, 1988. [On line]. Available: https://bit.ly/2SdqGyq

J. K. Finch, “The single-valued extension property on a Banach space”, Pacific journal of mathematics, vol. 58, no. 1, pp. 61-69, 1975. [On line]. Available: https://bit.ly/394Dazc

K. B. Laursen and M. M. Neumann, An introduction to local spectral theory, Oxford: Oxford University Press, 2000.

K. B. Laursen and P. Vrbová, “Some remarks on the surjectivity spectrum of linear operators”, Czechoslovak mathematical journal, vol. 39, no. 4, pp. 730-739, 1989. [On line]. Available: https://bit.ly/2tygLva

H. Lian and Q. Zeng, “An extension of Cline's formula for generalized Drazin inverse”, Turkish journal of mathematics, vol. 40, pp. 161-165, 2016, doi: 10.3906/mat-1505-4.

M. Mbekhta, “Généralisation de la décomposition de Kato aux opérateurs paranormaux et spectraux”, Glasgow mathematical journal, vol. 29, no. 2, pp. 159-175, Jul. 1987, doi: 10.1017/S0017089500006807.

M. Mbekhta, “Sur la théorie spectrale locale et limite des nilpotents”, Proceeding of the. american mathematical society, vol. 110, no. 3 pp. 621-631, 1990, doi: 10.1090/S0002-9939-1990-1004421-1.

T. L. Miller, V. G. Miller, and M. M. Neumann, “Localization in the spectral theory of operators on banach spaces,” in Function spaces, vol. 328, K. Jarosz, Ed. Providence, RI: American mathematical society, 2003, pp. 247–262, doi: 10.1090/conm/328.

T. L. Miller, V. G. Miller and M. M. Neumann, “On operators with closed analytic core”, Rendiconti del circolo matematico di Palermo, vol. 51, no. 3, pp. 495-502, Oct. 2002, doi: 10.1007/BF02871857.

V. G. Miller and H. Zguitti, “New extensions of Jacobson's lemma and Cline's formula”, Rendiconti del circolo matematico di Palermo, vol. 67,no. 1, pp. 105-114, Apr. 2018, doi: 10.1007/s12215-017-0298-6.

V. Müller, Spectral theory of linear operators: and spectral systems in Banach algebras, vol. 139. Basel: Birkhäuser, 2007, doi: 10.1007/978-3-7643-8265-0.

K. Yan and X. C. Fang, “Common properties of the operator products in local spectral theory”, Acta mathematica sinica, english series, vol. 31, no. 11, pp. 1715-1724, Nov. 2015, doi: 10.1007/s10114-015-5116-5.

Q.P. Zeng, H.J. Zhong, “Common properties of bounded linear operators AC and BA: spectral theory”, Mathematische nachrichte, vol. 287, no. 5-6, pp. 717-725, Apr. 2014, doi: 10.1002/mana.201300123.

Q. P. Zeng and H. J. Zhong, “Common properties of bounded linear operators AC and BA: local spectral theory”, Journal of mathematical analysis and applications, vol. 414, no. 2, pp. 553-560, Jun. 2014, doi: 10.1016/j.jmaa.2014.01.021.

E. H. Zerouali and H. Zguitti, “On the weak decomposition property (δw)”, Studia mathematica, vol. 167, no. 1, pp. 17-28, 2005. [On line]. Available: https://bit.ly/2vK5Y1r

Published
2020-02-04
How to Cite
[1]
H. Zguitti, “Further common local spectral properties for bounded linear operators”, Proyecciones (Antofagasta, On line), vol. 39, no. 1, pp. 243-259, Feb. 2020.
Section
Artículos