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Generalized spectrum and commuting compact perturbations

Published online by Cambridge University Press:  20 January 2009

Vladimir Rakočević
Vladimir Rakočević
Affiliation:
University of Niš, Faculty of Philosophy, Department of Mathematics, Ćirila and Metodija 2, 18000 Niš, Yugoslavia
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Abstract

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Let X be an infinite-dimensional complex Banach space and denote the set of bounded (compact) linear operators on X by B(X) (K(X)). Let N(A) and R(A) denote, respectively, the null space and the range space of an element A of B(X). Set R(A∞)=∩nR(An) and k(A)=dim N(A)/(N(A)∩R(A∞)). Let σg(A) = ℂ\{λ∈ℂ:R(A−λ) is closed and k(A−λ)=0} denote the generalized (regular) spectrum of A. In this paper we study the subset σgb(A) of σg(A) defined by σgb(A) = ℂ\{λ∈ℂ:R(A−λ) is closed and k(A−λ)<∞}. Among other things, we prove that if f is a function analytic in a neighborhood of σ(A), then σgb(f(A)) = f(σgb(A)).

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 36 , Issue 2 , June 1993 , pp. 197 - 209
Copyright
Copyright © Edinburgh Mathematical Society 1993

References

REFERENCES

1.Apostol, C., The reduced minimum modulus, Michigan Math. J. 32 (1985), 279–294.CrossRefGoogle Scholar
2.Förster, K.-H. and Kaashoek, M. A., The asymptotic behaviour of the reduced minimum modulus of a Fredholm operator, Proc. Amer. Math. Soc. 49 (1975), 123–131.Google Scholar
3.Goldman, M. A., On the stability of normal solvability of linear equations (Russian), Dokl. Akad. Nauk SSSR. 100 (1955), 201–204.Google Scholar
4.González, M., Null spaces and ranges of polynomials of operators, Publ. Mat. 32 (1988), 167–170.CrossRefGoogle Scholar
5.Grabiner, S., Uniform ascent and descent of bounded operators, J. Math. Soc. Japan 34 (1982), 317–337.CrossRefGoogle Scholar
6.Kaashoek, M. A., Stability theorems for closed linear operators, Indag. Math. 27 (1965), 452–466.CrossRefGoogle Scholar
7.Kaashoek, M. A., Ascent, descent, nullity and defect, a note on a paper by A. E. Taylor, Math. Ann. 172 (1967), 105–115.CrossRefGoogle Scholar
8.Krachkovskii, S. N., On the extended region of the singularity of the operator Tλ = E–λA (Russian), Dokl. Akad. Nauk SSSR 96 (1954), 1101–1104.Google Scholar
9.Kato, T., Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Analyse Math. 6 (1958), 261–322.CrossRefGoogle Scholar
10.Labrousse, J. P., Les opérateurs quasi-Fredholm: Une généralisation des opérateurs semi-Fredholm, Rend. Circ. Mat. Palermo (2) 29 (1980), 161–258.CrossRefGoogle Scholar
11.Lay, D. C., Spectral analysis using ascent, descent, nullity and defect, Math. Ann. 184 (1970), 197–214.CrossRefGoogle Scholar
12.Markus, A. S., On one theorem of F. Reisz (Russian), Uch. Zap. Kishinev. Gos. Univ. 17 (1955), 73–76.Google Scholar
13.Mbekhta, M., Résolvant généralisé et théorie spectrale, J. Operator Theory 21 (1989), 69–105.Google Scholar
14.Nikolskh, S. M., Linear equations in normed linear spaces (Russian), Bull. Acad. Sci. URSS Ser. Math. 7 (1943), 147–166.Google Scholar
15.Oliver, R. K., Note on a duality relation of Kaashoek, Indag. Math. 28 (1966), 364–368.CrossRefGoogle Scholar
16.Searcóid, M. Ó and West, T. T., Continuity of the generalized kernel and range of semi-Fredholm operators, Math. Proc. Cambridge Philos. Soc. 105 (1989), 513–522.CrossRefGoogle Scholar
17.Rakočević, V., Approximate point spectrum and commuting compact perturbations, Glasgow Math. J. 28 (1986), 193–198.CrossRefGoogle Scholar
18.Rakočević, V. and Zemánek, J., Lower s-numbers and their asymptotic behaviour, Studia Math. 91 (1988), 231–239.CrossRefGoogle Scholar
19.West, T. T., A Riesz-Schauder theorem for semi-Fredholm operators, Proc. Roy. Irish Acad. Sect. A 87 (1987), 137–146.Google Scholar
20.Zemánek, J., Geometric characteristics of semi-Friedholm operators and their asymptotic behaviour, Studia Math. 80 (1984), 219–234.CrossRefGoogle Scholar
21.Zemánek, J., Compressions and the Weyl-Browder spectra, Proc. Roy. Irish Acad. Sect. A 86 (1986), 57–62.Google Scholar
22.Zemánek, J., The reduced minimum modulus and the spectrum, Integral Equations Operator Theory 12 (1989), 449–454.CrossRefGoogle Scholar