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Proyecciones (Antofagasta)

Print version ISSN 0716-0917

Proyecciones (Antofagasta) vol.39 no.1 Antofagasta Feb. 2020 


Furter common local spectral properties for bounded linear operators

1 Sidi Mohamed Ben Abdellah University, Dept. of Mathematics, Fez, Morocco. E-mail:


We study common local spectral properties for bounded linear operators A ∈ ℒ(X,Y)

and B,C ∈ ℒ (Y,X) such that


We prove that AC and BA share the single valued extension property, the Bishop property (β), the property (β ε ), the decomposition property (δ) and decomposability. Closedness of analytic core and quasinilpotent part are also investigated. Some applications to Fredholm operators are given.

Keywords: Jacobson's lemma; Common properties; Local spectral theory.

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

[2] 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. [ Links ]

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

[4] 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. [ Links ]

[5] 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. [ Links ]

[6] 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. [ Links ]

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

[8] 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. [ Links ]

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

[10] 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: ]

[11] 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: ]

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

[13] 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: ]

[14] 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. [ Links ]

[15] 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. [ Links ]

[16] 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. [ Links ]

[17] T. L. Miller, V. G. Miller, andM. 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. [ Links ]

[18] T. L. Miller , V. G. Miller andM. 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. [ Links ]

[19] 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. [ Links ]

[20] 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. [ Links ]

[21] 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. [ Links ]

[22] 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. [ Links ]

[23] 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. [ Links ]

[24] E. H. Zerouali andH. Zguitti , “On the weak decomposition property (δw)”, Studia mathematica, vol. 167, no. 1, pp. 17-28, 2005. [On line]. Available: ]

Received: March 2019; Accepted: April 2019

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