TRACE EXPANSIONS AND THE NONCOMMUTATIVE RESIDUE FOR MANIFOLDS WITH BOUNDARY

For a pseudodifferential boundary operator A of order � 2 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+� j)/2, j 2 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 �) k ) in powersl/2 and log-powersl/2 log �, 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). then is a holomorphic function of s for large Re s. We show that it extends to a mero- morphic function on the whole complex plane with at most double poles. Moreover, we prove that the noncommutative residue res(A) of A can be recovered as a residue in this