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A resolvent approach to traces and zeta Laurent expansions. (English) Zbl 1073.58021

Booß-Bavnbek, Bernhelm (ed.) et al., Spectral geometry of manifolds with boundary and decomposition of manifolds. Proceedings of the workshop, Roskilde, Denmark, August 6–9, 2003. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3536-X/pbk). Contemporary Mathematics 366, 67-93 (2005).
Summary: Classical pseudodifferential operators \(A\) on closed manifolds are considered. It is shown that the basic properties of the canonical trace \(\text{TR}\,A\) introduced by Kontsevich and Vishik are easily proven by identifying it with the leading nonlocal coefficient \(C_0 (A,P)\) in the trace expansion of \(A(P-\lambda)^{-N}\) (with an auxiliary elliptic operator \(P)\), as determined in a joint work with Seeley 1995. The definition of \(\text{TR}\,A\) 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 \(\zeta(A,P,s)=\text{Tr}(AP^{-s})\), extended meromorphically to \(\mathbb{C}\), \(C_0(A,P)\) equals the coefficient of \(s_0\) in the Laurent expansion at \(s=0\) when \(P\) is invertible. In the mentioned parity cases, \(\zeta(A,P,s)\) is regular at all integer points. The higher Laurent coefficients \(C_j(A,P)\) at \(s=0\) are described as leading nonlocal coefficients \(C_0(B,P)\) in trace expansions of resolvent expressions \(B(P-\lambda)^{-N}\), 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 \(\text{TR}\,B\) in restricted cases, and the formula of Lesch for \(\text{TR}\,B\) in terms of a finite part integral of the symbol is extended to the parity cases.
For the entire collection see [Zbl 1057.58001].

MSC:

58J42 Noncommutative global analysis, noncommutative residues
35S05 Pseudodifferential operators as generalizations of partial differential operators
58J35 Heat and other parabolic equation methods for PDEs on manifolds
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
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