The sectorial projection defined from logarithms

Abstract

For a classical elliptic pseudodifferential operator P of order m>0 on a closed manifold X, such that the eigenvalues of the principal symbol p_m(x,\xi) have arguments in \,]\theta,\phi [\, and \,]\phi, \theta +2\pi [\, (\theta <\phi <\theta +2\pi), the sectorial projection \Pi_{\theta, \phi}(P) is defined essentially as the integral of the resolvent along {e^{i\phi}R_+}\cup {e^{i\theta}R_+}. In a recent paper, Booss-Bavnbek, Chen, Lesch and Zhu have pointed out that there is a flaw in several published proofs that \P_{\theta, \phi}(P) is a \psi do of order 0; namely that p_m(x,\xi) cannot in general be modified to allow integration of (p_m(x,\xi)-\lambda)^{-1} along {e^{i\phi}R_+}\cup {e^{i\theta}R_+} simultaneously for all \xi . We show that the structure of \Pi_{\theta, \phi}(P) as a \psi do of order 0 can be deduced from the formula \Pi_{\theta, \phi}(P)= (i/(2\pi))(\log_\theta (P) - \log_\phi (P)) proved in an earlier work (coauthored with Gaarde). In the analysis of \log_\theta (P) one need only modify p_m(x,\xi) in a neighborhood of e^{i\theta}R_+; this is known to be possible from Seeley's 1967 work on complex powers.