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Index of B-Fredholm operators and generalization of a Weyl theorem

Author: M. Berkani
Journal: Proc. Amer. Math. Soc. 130 (2002), 1717-1723
MSC (1991): Primary 47A53, 47A55
Published electronically: October 17, 2001
MathSciNet review: 1887019
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Abstract: The aim of this paper is to show that if $S$ and $T$ are commuting B-Fredholm operators acting on a Banach space $X$, then $ST$ is a B-Fredholm operator and $ind(ST)=ind(S)+ind(T)$, where $ind$ means the index. Moreover if $T$ is a B-Fredholm operator and $F$ is a finite rank operator, then $T+F$ is a B-Fredholm operator and $ind(T+F)= ind(T).$ We also show that if $0$ is isolated in the spectrum of $T$, then $T$ is a B-Fredholm operator of index $0$ if and only if $T$ is Drazin invertible. In the case of a normal bounded linear operator $T$ acting on a Hilbert space $H$, we obtain a generalization of a classical Weyl theorem.

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  • 1. B. A. Barnes, Riesz Points and Weyl's Theorem. Integr. Equ. Oper. Theory 34 (1999) 187-196. MR 2000d:47006
  • 2. M. Berkani, On a class of quasi-Fredholm operators. Integr. Equ. Oper. Theory 34 (1999), 244-249. MR 2000d:47023
  • 3. M. Berkani, Restriction of an operator to the range of its powers.. Studia Mathematica, Vol 140(2), (2000), 163-175. MR 2001g:47021
  • 4. M. Berkani, M. Sarih, On semi-B-Fredholm operators. To appear in Glasgow Mathematical Journal.
  • 5. M. Berkani, M. Sarih, An Atkinson-type theorem for B-Fredholm operators, To appear in Studia Mathematica.
  • 6. M. P. Drazin, Pseudoinverse in associative rings and semigroups. Amer. Math. Monthly 65 (1958), 506-514. MR 20:5217
  • 7. S. Grabiner, Uniform ascent and descent of bounded operators; J. Math. Soc. Japan 34, No. 2 (1982), 317-337. MR 84a:47003
  • 8. R. Harte, Invertibility and singularity for bounded linear operators; Marcel Dekker. New York, Basel, 1988. MR 90a:15019
  • 9. M. Kaashoek, Ascent, Descent, Nullity and Defect, a Note on a Paper by A.E. Taylor; Math. Annalen 172, 105-115 (1967). MR 36:5719
  • 10. J. J. Koliha, A generalized Drazin inverse, Glasgow Math. J. 38 (1996), 367-381. MR 98b:46065
  • 11. J. P. Labrousse, Les opérateurs quasi-Fredholm: une généralisation des opérateurs semi-Fredholm; Rend. Circ. Math. Palermo (2), 29 (1980), 161-258. MR 83c:47022
  • 12. D. C. Lay, Spectral analysis using ascent, descent, nullity and defect; Math. Ann. 184, 197-214 (1970).
  • 13. D. C. Lay, A. E. Taylor, Introduction to Functional Analysis, Second edition, Wiley, New York, 1980. MR 81b:46001
  • 14. S. Roch, B. Silbermann, Continuity of generalized inverses in Banach algebras. Studia Mathematica 136 (3), (1999), 197-227. MR 2000m:46101
  • 15. H. Weyl, Über beschränkte quadratische Formen, deren Differenz vollstetig ist. Rend. Circ. Mat. Palermo 27 (1909), 373-392.

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Additional Information

M. Berkani
Affiliation: Département de Mathématiques, Faculté des Sciences, Université Mohammed I, Oujda, Maroc

Received by editor(s): December 5, 2000
Published electronically: October 17, 2001
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2001 American Mathematical Society

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