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 $\mathsf{R}_{-}$. 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 $\zeta (A,B,s)= {\operatorname {Tr}}(AB^{-s})$ (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 $\theta <\varphi$, one can define the sectorial projection $\Pi_{\theta,\varphi} (B)$ whose range contains the generalized eigenspaces for eigenvalues with argument in $\left]\theta,\varphi \right[$; this is studied in the last part of the paper. The operator $\Pi_{\theta ,\varphi}(B)$ is shown to be proportional to the difference between $\log_{\theta}B$ and $\log_{\varphi} B$, 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.