On the isolated points of the surjective spectrum of a bounded operator
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- by Manuel González, Mostafa Mbekhta and Mourad Oudghiri PDF
- Proc. Amer. Math. Soc. 136 (2008), 3521-3528 Request permission
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
- Pietro Aiena, Maria Luisa Colasante, and Manuel González, Operators which have a closed quasi-nilpotent part, Proc. Amer. Math. Soc. 130 (2002), no. 9, 2701–2710. MR 1900878, DOI 10.1090/S0002-9939-02-06386-4
- Pietro Aiena, T. Len Miller, and Michael M. Neumann, On a localised single-valued extension property, Math. Proc. R. Ir. Acad. 104A (2004), no. 1, 17–34. MR 2139507, DOI 10.3318/PRIA.2004.104.1.17
- Ion Colojoară and Ciprian Foiaş, Theory of generalized spectral operators, Mathematics and its Applications, Vol. 9, Gordon and Breach Science Publishers, New York-London-Paris, 1968. MR 0394282
- Weibang Gong and Libin Wang, Mbekhta’s subspaces and a spectral theory of compact operators, Proc. Amer. Math. Soc. 131 (2003), no. 2, 587–592. MR 1933350, DOI 10.1090/S0002-9939-02-06639-X
- Domingo A. Herrero, Approximation of Hilbert space operators. Vol. 1, 2nd ed., Pitman Research Notes in Mathematics Series, vol. 224, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989. MR 1088255
- J. J. Koliha, Isolated spectral points, Proc. Amer. Math. Soc. 124 (1996), no. 11, 3417–3424. MR 1342031, DOI 10.1090/S0002-9939-96-03449-1
- Kjeld B. Laursen and Michael M. Neumann, An introduction to local spectral theory, London Mathematical Society Monographs. New Series, vol. 20, The Clarendon Press, Oxford University Press, New York, 2000. MR 1747914
- Mostafa Mbekhta, Généralisation de la décomposition de Kato aux opérateurs paranormaux et spectraux, Glasgow Math. J. 29 (1987), no. 2, 159–175 (French). MR 901662, DOI 10.1017/S0017089500006807
- Mostafa Mbekhta, Sur la théorie spectrale locale et limite des nilpotents, Proc. Amer. Math. Soc. 110 (1990), no. 3, 621–631 (French). MR 1004421, DOI 10.1090/S0002-9939-1990-1004421-1
- Mostafa Mbekhta, On the generalized resolvent in Banach spaces, J. Math. Anal. Appl. 189 (1995), no. 2, 362–377. MR 1312050, DOI 10.1006/jmaa.1995.1024
- M. Mbekhta and A. Ouahab, Opérateur s-régulier dans un espace de Banach et théorie spectrale, Acta Sci. Math. (Szeged) 59 (1994), no. 3-4, 525–543 (French, with English summary). MR 1317171
- T. Len Miller, Vivien G. Miller, and Michael M. Neumann, On operators with closed analytic core, Rend. Circ. Mat. Palermo (2) 51 (2002), no. 3, 495–502. MR 1947470, DOI 10.1007/BF02871857
- Mourad Oudghiri, Weyl’s and Browder’s theorems for operators satisfying the SVEP, Studia Math. 163 (2004), no. 1, 85–101. MR 2047466, DOI 10.4064/sm163-1-5
- Christoph Schmoeger, On isolated points of the spectrum of a bounded linear operator, Proc. Amer. Math. Soc. 117 (1993), no. 3, 715–719. MR 1111438, DOI 10.1090/S0002-9939-1993-1111438-8
- Florian-Horia Vasilescu, Analytic functional calculus and spectral decompositions, Mathematics and its Applications (East European Series), vol. 1, D. Reidel Publishing Co., Dordrecht; Editura Academiei Republicii Socialiste România, Bucharest, 1982. Translated from the Romanian. MR 690957
Additional Information
- Manuel González
- Affiliation: Departamento de Matemáticas, Facultad de Ciencias, Universidad de Cantabria, E-39071 Santander, España
- MR Author ID: 219505
- Email: gonzalem@unican.es
- Mostafa Mbekhta
- Affiliation: Université de Lille I, UFR de Mathématiques, 59655 Villeneuve d’Ascq cedex, France
- MR Author ID: 121980
- Email: mostafa.mbekhta@math.univ-lille1.fr
- Mourad Oudghiri
- Affiliation: Département de Mathématiques et Informatique, Faculté des Sciences d’Oujda, Maroc
- Email: oudghiri@fso.ump.ma
- 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
- © Copyright 2008 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 136 (2008), 3521-3528
- MSC (2000): Primary 47A53; Secondary 47A68, 46B04
- DOI: https://doi.org/10.1090/S0002-9939-08-09549-X
- MathSciNet review: 2415036