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A Trace Formula for the Index of B-Fredholm Operators

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

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 ([email protected])
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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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