Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On the isolated points of the surjective spectrum of a bounded operator

Authors: Manuel González, Mostafa Mbekhta and Mourad Oudghiri
Journal: Proc. Amer. Math. Soc. 136 (2008), 3521-3528
MSC (2000): Primary 47A53; Secondary 47A68, 46B04
Published electronically: May 15, 2008
MathSciNet review: 2415036
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For a bounded operator $ T$ acting on a complex Banach space, we show that if $ T-\lambda$ is not surjective, then $ \lambda$ is an isolated point of the surjective spectrum $ \sigma_{su}(T)$ of $ T$ if and only if $ X=H_0(T-\lambda)+K(T-\lambda)$, where $ H_0(T)$ is the quasinilpotent part of $ T$ and $ K(T)$ is the analytic core for $ T$. Moreover, we study the operators for which $ \dim K(T) < \infty$. We show that for each of these operators $ T$, there exists a finite set $ E$ consisting of Riesz points for $ T$ such that $ 0\in \sigma (T)\setminus E$ and $ \sigma (T)\setminus E$ is connected, and derive some consequences.

References [Enhancements On Off] (What's this?)

  • 1. P. Aiena, M. L. Colasante and M. González, Operators which have a closed quasi-nilpotent part, Proc. Amer. Math. Soc. 130 (2002), 2701-2710. MR 1900878 (2003g:47008)
  • 2. P. Aiena, T. L. Miller and M. M. Neumann, On a localised single-valued extension property, Math. Proc. Irish Math. Soc. 104A (2004), 17-34. MR 2139507 (2005k:47011)
  • 3. I. Colojoara and C. Foias, Theory of generalized spectral operators, Gordon and Breach, New York, 1968. MR 0394282 (52:15085)
  • 4. W. Gong and L. Wang, Mbekhta's subspaces and a spectral theory of compact operators, Proc. Amer. Math. Soc. 131 (2003), 587-592. MR 1933350 (2003g:47004)
  • 5. A. A. Herrero, Approximation of Hilbert space operators. Vol. 1. Second edition. Pitman Research Notes in Mathematics Series, 224. Longman Scientific & Technical, Harlow, 1989. MR 1088255 (91k:47002)
  • 6. J. J. Koliha, Isolated spectral points, Proc. Amer. Math. Soc. 124 (1996), 3417-3424. MR 1342031 (97a:46057)
  • 7. K. B. Laursen and M. M. Neumann, An introduction to local spectral theory, London Math. Soc. Monog. New Series, 20. The Clarendon Press, Oxford Univ. Press, New York, 2000. MR 1747914 (2001k:47002)
  • 8. M. Mbekhta, Généralisation de la décomposition de Kato aux opérateurs paranormaux et spectraux, Glasgow Math. J. 29 (1987), 159-175. MR 901662 (88i:47010)
  • 9. M. Mbekhta, Sur la théorie spectrale locale et limite des nilpotents, Proc. Amer. Math. Soc. 110 (1990), 621-631. MR 1004421 (91b:47004)
  • 10. M. Mbekhta, On the generalized resolvent in Banach spaces, J. Math. Anal. Appl. 189 (1995), 362-377. MR 1312050 (96f:47004)
  • 11. M. Mbekhta and A. Ouahab, Opérateur s-régulier dans un espace de Banach et théorie spectrale, Acta Sci. Math. (Szeged) 59 (1994), 525-543. MR 1317171 (96a:47018)
  • 12. T. L. Miller, V. G. Miller and M. M. Neumann, On operators with closed analytic core, Rend. Circ. Mat. Palermo (2) 51 (2002), 495-502. MR 1947470 (2003h:47008)
  • 13. M. Oudghiri, Weyl's and Browder's theorems for operators satisfying the SVEP, Studia Math. 163 (2004), 85-101. MR 2047466 (2005b:47006)
  • 14. Ch. Schmoeger, On isolated points of the spectrum of a bounded linear operator, Proc. Amer. Math. Soc. 117 (1993), 715-719. MR 1111438 (93d:47007)
  • 15. F.-H. Vasilescu, Analytic functional calculus and spectral decomposition, Editura Academiei, Bucharest, 1982. MR 690957 (85b:47016)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47A53, 47A68, 46B04

Retrieve articles in all journals with MSC (2000): 47A53, 47A68, 46B04

Additional Information

Manuel González
Affiliation: Departamento de Matemáticas, Facultad de Ciencias, Universidad de Cantabria, E-39071 Santander, España

Mostafa Mbekhta
Affiliation: Université de Lille I, UFR de Mathématiques, 59655 Villeneuve d’Ascq cedex, France

Mourad Oudghiri
Affiliation: Département de Mathématiques et Informatique, Faculté des Sciences d’Oujda, Maroc

Keywords: Isolated points of the surjective spectrum, analytic core, quasinilpotent part of an operator.
Received by editor(s): July 2, 2007
Published electronically: May 15, 2008
Additional Notes: This research was partially supported by DGI (Spain), Proyecto MTM2007-67994.
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2008 American Mathematical Society

American Mathematical Society