A Resolvent Approach to Traces and Zeta Laurent Expansions

Classical pseudodifferential operators A on closed manifolds are considered. It is shown that the basic properties of the canonical trace TRA introduced by Kontsevich and Vishik are easily proved by identifying it with the leading nonlocal coefficient C0(A, P ) in the trace expansion of A(P−λ) (with an auxiliary elliptic operator P ), as determined in a joint work with Seeley 1995. The definition of TRA is extended from the cases of noninteger order, or integer order and even-even parity on odd-dimensional manifolds, to the case of even-odd parity on even-dimensional manifolds. For the generalized zeta function ζ(A,P, s) = Tr(AP), extended meromorphically to C, C0(A, P ) equals the coefficient of s in the Laurent expansion at s = 0 when P is invertible. In the mentioned parity cases, ζ(A,P, s) is regular at all integer points. The higher Laurent coefficients Cj(A, P ) at s = 0 are described as leading nonlocal coefficients C0(B, P ) in trace expansions of resolvent expressions B(P −λ) , with B log-polyhomogeneous as defined by Lesch (here −C1(I, P ) = C0(logP,P ) gives the zeta-determinant). C0(B, P ) is shown to be a quasi-trace in general, a canonical trace TRB in restricted cases, and the formula of Lesch for TRB in terms of a finite part integral of the symbol is extended to the parity cases.