A resolvent approach to traces and zeta Laurent expansions
| dc.creator | Grubb, Gerd | |
| dc.date | 2003-11-06 | |
| dc.date | 2005-12-19 | |
| dc.date.accessioned | 2026-07-07T06:35:47Z | |
| dc.date.available | 2026-07-07T06:35:47Z | |
| dc.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. | |
| dc.description | 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 | |
| dc.identifier | https://arxiv.org/abs/math/0311081 | |
| dc.identifier | http://arxiv.org/abs/math/0311081 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/99900 | |
| dc.subject | Analysis of PDEs | |
| dc.subject | Spectral Theory | |
| dc.subject | 58J42; 35S05; 58J35; 41A60 | |
| dc.title | A resolvent approach to traces and zeta Laurent expansions | |
| dc.type | text |