In this paper we take the investigation begun in [11] much further. We show how to generalize a formula known from the setting of the L2 index theorem to the non-Fredholm setting. Restricting back to the case of selfadjoint Fredholm operators our formula extends the result of [22] in the sense of relaxing the discrete spectrum condition. It also generalizes some other Fredholm operator results of , and that permit essential spectrum for the operators in the path. Our result may also apply however when the operators {D(t)} have essential spectrum equal to the whole real line.
Our main theorem gives a trace formula relating the homological index of [7] to an integral formula that is known, for a path of selfadjoint Fredholms with compact resolvent and with unitarily equivalent endpoints, to compute spectral flow. Our formula however, applies to paths of selfadjoint non-Fredholm operators. We interpret this as indicating there is a generalization of spectral flow to the non-Fredholm setting.