## ABSTRACT

With the APS boundary condition, the appearance in (In.6) of the eta invariant is a little less striking, since it represents just a part of the in formation in the projection Q+. The condition (In.8) reflects the squareintegrability of an extension of the solution into t < 0, i.e. to the whole of the original manifold X. It will not be encountered below1. As already noted, the invertibility of the operator 3+, which in general for an exact 6-metric is not quite as simple as (In.7), is attacked directly and its gener alized inverse and the associated heat kernels are shown to be elements of the appropriate space of 6-pseudodifferential operators. 4. Preliminaries to the proof.