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

González, Manuel; Mbekhta, Mostafa; Oudghiri, Mourad

doi:10.1090/S0002-9939-08-09549-X

On the isolated points of the surjective spectrum of a bounded operator
HTML articles powered by AMS MathViewer

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