Logarithms and sectorial projections for elliptic boundary problems

Abstract

On a compact manifold with boundary, consider the realization B of an elliptic, possibly pseudodifferential, boundary value problem having a spectral cut (a ray free of eigenvalues), say R_-. In the first part of the paper we define and discuss in detail the operator log B; its residue (generalizing the Wodzicki residue) is essentially proportional to the zeta function value at zero, zeta(B,0), and it enters in an important way in studies of composed zeta functions zeta(A,B,s)=Tr(AB^{-s}) (pursued elsewhere). There is a similar definition of the operator log_theta B, when the spectral cut is at a general angle theta. When B has spectral cuts at two angles theta < phi, one can define the sectorial projection Pi_{theta,phi}(B) whose range contains the generalized eigenspaces for eigenvalues with argument in ] theta, phi [; this is studied in the last part of the paper. The operator Pi_{theta,phi}(B) is shown to be proportional to the difference between log_theta B and log_phi B, having slightly better symbol properties than they have. We show by examples that it belongs to the Boutet de Monvel calculus in many special cases, but lies outside the calculus in general.