A trace formula for the index of B-Fredholm operators

TL;DR

This paper introduces a trace formula for the index of B-Fredholm operators in Banach algebras, extending Fedosov's formula and establishing a neighborhood theorem for the index.

Contribution

It defines B-Fredholm elements in Banach algebras modulo an ideal and derives a trace-based index formula extending Fedosov's result.

Findings

Index of B-Fredholm operators equals the trace of a commutator involving a Drazin inverse.

Established a punctured neighborhood theorem for the index in semi-simple Banach algebras.

Extended Fedosov's trace formula to B-Fredholm operators.

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 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, T_0],$ where $T_0$ 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.