Trace Expansions and the Noncommutative Residue for Manifolds with Boundary

For a pseudodifferential boundary operator A of integer order \nu and class zero (in the Boutet de Monvel calculus) on a compact n-dimensional manifold with boundary, we consider the function Trace(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 Trace(AB^{-s}) has a meromorphic extension to the complex plane with poles at the half-integers s = (n+\nu-j)/2, j = 0,1,... (possibly double for s<0), and we prove that its residue at zero 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 Trace(A(B-\lambda)^{-k}) in powers of \lambda^{-j/2} and log-powers \lambda^{-j/2} log \lambda, where the noncommutative residue equals the coefficient of the highest log-power. There is a related expansion for Trace(A exp(-tB)).