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.Comment: Quotations elaborated, 6 pages, to appear in Mathematica Scandinavic
