TY - BOOK

T1 - Projections and residues on manifolds with boundary

AU - Gaarde, Anders Borg

PY - 2008

Y1 - 2008

N2 - 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.

AB - 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.

M3 - Ph.D. thesis

SN - 978-87-91927-31-7

BT - Projections and residues on manifolds with boundary

CY - København

ER -