A resolvent approach to traces and zeta Laurent expansions
Abstract
Description
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.
Updated with known corrections. The paper has appeared in AMS Contemporary Math. Proceedings, vol. 366 "Spectral Geometry of Manifolds with Boundary and Decomposition of Manifolds", 2005, pp. 67-93
Updated with known corrections. The paper has appeared in AMS Contemporary Math. Proceedings, vol. 366 "Spectral Geometry of Manifolds with Boundary and Decomposition of Manifolds", 2005, pp. 67-93