Trace expansions and the noncommutative residue for manifolds with boundary

For a pseudodifferential boundary operator A of order v epsilon Z and class 0 (in the Boutet do Monvel calculus) on a compact n-dimensional manifold w ith boundary, we consider the function Tr(AB(-s)), where B is an auxiliary system formed of the Dirichlet realization of a second order strongly ellip tic differential operator and an elliptic operator on the boundary. We prov e that Tr(AB(-s)) has a meromorphic extension to C with poles at the half-i ntegers s = (n + v - j)/2, j epsilon N (possibly double for s < 0), and we prove that its residue at 0 equals the noncommutative residue of A, as defi ned by Fedosov, Golse, Leichtnam and Schrohe by a different method. To achi eve this, we establish a full asymptotic expansion of Tr (A(B - <lambda>)(- k)) in powers lambda (-1/2) and log-powers lambda (-1/2) log lambda where t he noncommutative residue equals the coefficient of the highest order log-p ower. There is a related expansion of Tr(Ae(-tB)).