Traces and Quasi-traces on the Boutet de Monvel Algebra
Abstract
Description
We construct an analogue of Kontsevich and Vishik's canonical trace for a class of pseudodifferential boundary value problems in Boutet de Monvel's calculus on compact manifolds with boundary.
For an operator A in the calculus (of class zero), and an auxiliary operator B, formed of the Dirichlet realization of a strongly elliptic second-order differential operator and an elliptic operator on the boundary, we consider the coefficient C_0(A,B) of (-λ)^{-N} in the asymptotic expansion of the resolvent trace Tr(A(B-λ)^{-N}) (with N large) in powers and log-powers of λas λtends to infinity in a suitable sector of the complex plane. C_0(A,B) identifies with the coefficient of s^0 in the Laurent series for the meromorphic extension of the generalized zeta function ζ(A,B,s)= Tr(AB^{-s}) at s=0, when B is invertible.
We show that C_0(A,B) is in general a quasi-trace, in the sense that it vanishes on commutators [A,A'] modulo local terms, and has a specific value independent of B modulo local terms; and we single out particular cases where the local ``errors'' vanish so that C_0(A,B) is a well-defined trace of A.
Our main tool is a precise analysis of the asymptotic expansion of the resolvent trace, based on pseudodifferential calculations involving rational functions (in particular Laguerre functions) of the normal variable.
Final version to appear in Ann. Inst. Fourier
Final version to appear in Ann. Inst. Fourier