It is a well-known result that the noncommutative residue of a pseudodifferential projection is zero on a compact manifold without boundary. Equivalently, the value of the zeta-function of P at zero, ¿_¿(P, 0), is independent of ¿ for any elliptic operator P. Here ¿ denotes the angle of a ray where the resolvent of P has minimal growth.

In this thesis, we consider the analogous questions on a compact manifold with boundary. We show that the noncommutative residue is zero for any projection in Boutet de Monvel’s calculus of pseudodifferential boundary problems.

For an elliptic boundary problem {P₊ + G, T }, with the corresponding realization B = (P + G)_T, we de¿ne the sectorial projection ¿_¿,¿(B) and the residue of this projection. We discuss whether this residue is always zero, through various analyses of the structure of the pro jection. The question is interesting since ¿_¿(B, 0) is independent of ¿ exactly when the residues of the corresponding sectorial projections are zero; in particular this holds when the projections are in Boutet de Monvel’s calculus. This happens in certain cases, but we also give examples where the projections lie outside
the calculus.