2003/11/06 by Gerd Grubb, Grubb, Gerd
Mathematics · #35S05 #41A60 #58J35 #58J42 #Advanced Operator Algebra Research #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math.AP #math.SP #msc:35S05 #msc:41A60 #msc:58J35 #msc:58J42
paper · pdf · doi:10.48550/arxiv.math/0311081
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
openalex publication_date 2003/11/06 · arxiv created 2005/12/19 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 C0(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, C0(A,P) equals the coefficient of s0 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 coeficients C0(B,P) in trace expansions of resolvent expressions B(P-λ)-N, with B log-polyhomogeneous as defined by Lesch (here -C1(I,P)=C0(log P,P) gives the zeta-determinant). C0(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.