Trace Expansions and the Noncommutative Residue for Manifolds with Boundary

For a pseudodifferential boundary operator A of order ν ∈ Z and class 0 (in the Boutet de Monvel calculus) on a compact n-dimensional manifold with boundary, we consider the function Tr(AB−s), where B is an auxiliary system formed of the Dirichlet realization of a second order strongly elliptic differential operator and an elliptic operator on the boundary. We prove that Tr(AB−s) has a meromorphic extension to C with poles at the half-integers s = (n+ ν− j)/2, j ∈ N (possibly double for s < 0), and we prove that its residue at 0 equals the noncommutative residue of A, as defined by Fedosov, Golse, Leichtnam and Schrohe by a different method. To achieve this, we establish a full asymptotic expansion of Tr(A(B−λ)−k) in powers λ−l/2 and log-powers λ−l/2 log λ, where the noncommutative residue equals the coefficient of the highest order log-power. There is a related expansion of Tr(Ae−tB). The paper will appear in Journal Reine Angew. Math. (Crelle’s Journal).