Eta-invariant for parameter-dependent families with periodic coefficients

Abstract

On a closed smooth manifold, we consider operator families being linear combinations of parameter-dependent pseudodifferential operators with periodic coefficients. Such families arise in studying nonlocal elliptic problems on manifolds with isolated singularities and/or with cylindrical ends. The aim of the work is to construct the $\eta$-invariant for invertible families and to study its properties. We follow Melrose's approach who treated the $\eta$-invariant as a generalization of the winding number being equal to the integral the trace of the logarithmic derivative of the family. At the same time, the Melrose $\eta$-invariant is equal to the regularized integral of the regularized trace of the logarithmic derivative of the family. In our situation, for the trace regularization, we employ the operator of difference differentiating instead of the usual differentation used by Melrose. The main technical result is the fact that the operator of difference differentiation is an isomorphism between the spaces of functions with conormal asymptotics at infinity and this allows us to determine the regularized trace. Since the obtained regularized trace can increase at infinity, we also introduce a regularization for the integral. Our integral regularization involves an averaging operation. Then we establish the main properties of the $\eta$-invariant. Namely, the $\eta$-invariant in the sense of this work satisfies the logarithmic property and is a generalization of Melrose's $\eta$-invariant, that is, it coincides with it for usual parameter-dependent pseudodifferential operators. Finally, we provide a formula for the variation of the $\eta$-invariant under a variation of the family.