The sectorial projection defined from logarithms

TL;DR

This paper clarifies the structure of sectorial projections for elliptic pseudodifferential operators by relating them to differences of logarithms, addressing previous proof flaws and building on Seeley's work on complex powers.

Contribution

It demonstrates that sectorial projections can be expressed as differences of logarithms, resolving earlier issues with their classification as pseudodifferential operators.

Findings

Sectorial projection equals (i/2π) times the difference of logarithms.

Modification of principal symbol near specific rays suffices for analysis.

Addresses and corrects previous proof flaws in the literature.

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.