2016/09/06 by Berkani, Mohammed
#FOS: Mathematics #Operator Algebras (math.OA) #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.1609.01695
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 generate 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 a Fedosov's trace formula for Fredholm operators. In the case of a semi-simple Banach algebra, we prove a punctured neighborhood theorem for the index.