The sectorial projection defined from logarithms

Abstract

For a classical elliptic pseudodifferential operator $P$ of order $>0$ on a closed manifold $X$, such that the eigenvalues of the principal symbol $p_m(x,\xi)$ have arguments in $]\theta ,\varphi [$ and $]\varphi ,\theta +2\pi [$ ($\theta <\varphi <\theta +2\pi$), the sectorial projection $\Pi_{\theta,\varphi}(P)$ is defined essentially as the integral of the resolvent along $e^{i\varphi}\overline{\mathrm R}_{+}\cup e^{i\theta}\overline{\mathrm R}_{+}$. In a recent paper, Booss-Bavnbek, Chen, Lesch and Zhu have pointed out that there is a flaw in several published proofs that $\Pi_{\theta,\varphi}(P)$ 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 $e^{i\varphi}\overline{\mathrm R}_{+}\cup e^{i\theta}\overline{\mathrm R}_{+}$ simultaneously for all $\xi$. We show that the structure of $\Pi_{\theta,\varphi}(P)$ as a $\psi$do of order 0 can be deduced from the formula $\Pi_{\theta,\varphi}(P)=\frac {i}{2\pi}(\log_{\theta} P - \log_{\varphi} P)$ 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 $e^{i\theta}\overline{\mathrm R}_{+}$ this is known to be possible from Seeley's 1967 work on complex powers.