A resolvent approach to traces and zeta Laurent expansions

Abstract

Classical pseudodifferential operators A on closed manifolds are considered. It is shown that the basic properties of the canonical trace TR A introduced by Kontsevich and Vishik are easily proved by identifying it with the leading nonlocal coefficient C_0(A,P) in the trace expansion of A(P-λ)^{-N} (with an auxiliary elliptic operator P), as determined in a joint work with Seeley 1995. The definition of TR A 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^{-s}), extended meromorphically to C, C_0(A,P) equals the coefficient of s^0 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 C_j(A,P) at s=0 are described as leading nonlocal coeficients C_0(B,P) in trace expansions of resolvent expressions B(P-λ)^{-N}, with B log-polyhomogeneous as defined by Lesch (here -C_1(I,P)=C_0(\log P,P) gives the zeta-determinant). C_0(B,P) is shown to be a quasi-trace in general, a canonical trace TR B in restricted cases, and the formula of Lesch for TR B in terms of a finite part integral of the symbol is extended to the parity cases.