DIXMIERS TRACE FOR BOUNDARY-VALUE-PROBLEMS

Let X be a smooth manifold with boundary of dimension n > 1. The opera tors of order -n and type zero in Boutet de Monvel's calculus form a s ubset of Dixmier's trace ideal L-1,L-infinity(H) for the Hilbert space H = L-2(X, E) + L-2(partial derivative X, F) of L-2 sections in vecto r bundles E over X, F over partial derivative X. We show that, on thes e operators, Dixmier's trace can be computed in terms of the same expr essions that determine the noncommutative residue. In particular it is independent of the averaging procedure. However, the noncommutative r esidue and Dixmier's trace are not multiples of each other unless the boundary is empty. As a corollary we show how to compute Dixmier's tra ce for parametrices or inverses of classical elliptic boundary value p roblems of the form Pu = f; Tu = 0 with an elliptic differential opera tor P of order n in the interior and a trace operator T. In this parti cular situation, Dixmier's trace and the noncommutative residue do coi ncide up to a factor.