Traces and Quasi-traces on the Boutet De Monvel Algebra

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 C0(A,B) of (−λ) in the asymptotic expansion of the resolvent trace Tr(A(B − λ) ) (with N large) in powers and log-powers of λ for λ → ∞ in a suitable sector of C. C0(A,B) identifies with the coefficient of s in the Laurent series for the meromorphic extension of the generalized zeta function ζ(A,B, s) = Tr(AB) at s = 0, when B is invertible. We show that C0(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 C0(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.