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A Trace Formula for the Index of B-Fredholm Operators

Part of: Topological algebras, normed rings and algebras, Banach algebras General theory of linear operators

Published online by Cambridge University Press:  06 August 2018

Mohammed Berkani
Mohammed Berkani*
Affiliation:
Department of Mathematics, Science faculty of Oujda, University Mohammed I, Laboratory LAGA, Morocco (berkanimo@aim.com)
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Abstract

In this paper we define B-Fredholm elements in a Banach algebra A modulo an ideal J of A. When a trace function is given on the ideal J, it generates an index for B-Fredholm elements. In the case of a B-Fredholm operator T acting on a Banach space, we prove that its usual index ind(T) is equal to the trace of the commutator [T, T0], where T0 is a Drazin inverse of T modulo the ideal of finite rank operators, extending Fedosov's trace formula for Fredholm operators (see Böttcher and Silbermann [Analysis of Toeplitz operators, 2nd edn (Springer, 2006)]. In the case of a primitive Banach algebra, we prove a punctured neighbourhood theorem for the index.

Keywords

B-FredholmBanach algebraindextrace

MSC classification

Primary: 47A53: (Semi-) Fredholm operators; index theories
Secondary: 46H05: General theory of topological algebras
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 4 , November 2018 , pp. 1063 - 1068
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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References

1.Aupetit, B. and Mouton, H. du T., Trace and determinant in Banach algebras, Studia Math. 121 (1996), 115–136.Google Scholar
2.Barnes, B., Murphy, G. J., Smyth, M. R. F. and West, T. T., Riesz and Fredholm theory in Banach algebras (Pitman Publishing, 1982).Google Scholar
3.Berkani, M., On a class of quasi-Fredholm operators, Integr. Equ. Oper. Theory 34 (1999), 244–249.Google Scholar
4.Berkani, M., B-Fredholm elements in rings and algebras, Publ. Math. Debrecen. 92(1–2) (2018), 171–181.Google Scholar
5.Berkani, M. and Medkova, D., A note on the index of B-Fredholm operators, Math. Bohem. 129(2) (2004), 177–180.Google Scholar
6.Berkani, M. and Sarih, M., An Atkinson type theorem for B-Fredholm operators, Studia Math. 148 (2001), 251–257.Google Scholar
7.Böttcher, A. and Silbermann, B., Analysis of Toeplitz operators, 2nd edn (Springer, 2006).Google Scholar
8.Cvetković, M. D., Boasso, E. and Č, S.. Živković-Zlatanović, Generalized B-Fredholm Banach algebra elements, Med. J. Math. 13(5) (2016), 3729–3746.Google Scholar
9.Grobler, J. J. and Raubenheimer, H., The index for Fredholm elements in a Banach algebra via a a trace, Studia Math. 187(3) (2008), 281–297.Google Scholar
10.Murphy, G. J., Fredholm index theory and the trace, Proc. R. Irish Acad. 94A (1994), 161–166.Google Scholar