A Trace Formula for the Index of B-Fredholm Operators

Abstract

AbstractIn this paper we define B-Fredholm elements in a Banach algebraAmodulo an idealJofA. When a trace function is given on the idealJ, it generates an index for B-Fredholm elements. In the case of a B-Fredholm operatorTacting on a Banach space, we prove that its usual index ind(T) is equal to the trace of the commutator [T, T0], whereT0is a Drazin inverse ofTmodulo 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.