Noncommutative Residues, Dixmier's Trace, and Heat Trace Expansions on Manifolds with Boundary

For manifolds with boundary, we define an extension of Wodzicki's noncommutative residue to boundary value problems in Boutet de Monvel's calculus. We show that this residue can be recovered with the help of heat kernel expansions and explore its relation to Dixmier's trace.