A Trace Formula for the Index of B-Fredholm Operators. Issue 4 (6th November 2018)

• Record Type: Journal Article
• Authors: Berkani, Mohammed
• Is Part Of: Proceedings of the Edinburgh Mathematical Society. Volume 61:Issue 4(2018)
• Page Start: 1063
• Page End: 1068
• Publication Date: 2018-11-06
• Subjects: B-Fredholm -- Banach algebra -- index -- trace; Primary 47A53 -- 46H05; Mathematics -- Periodicals; 510
• DOI: 10.1017/S0013091518000147
• Languages: English
• ISSNs: 0013-0915

Abstract

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, T 0 ], where T 0 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.