The sectorial projection defined from logarithms - Grubb, Gerd (MATHEMATICA SCANDINAVICA, 2012)
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,ξ) have arguments in ]θ ,φ [ and ]φ ,θ +2π [ (θ <φ <θ +2π), the sectorial projection Π_θ,φ(P) is defined essentially as the integral of the resolvent along e^iφR_+e^iθR_+. In a recent paper, Booss-Bavnbek, Chen, Lesch and Zhu have pointed out that there is a flaw in several published proofs that Π_θ,φ(P) is a ψdo of order 0; namely that p_m(x,ξ) cannot in general be modified to allow integration of (p_m(x,ξ )-λ)^-1 along e^iφR_+e^iθR_+ simultaneously for all ξ. We show that the structure of Π_θ,φ(P) as a ψdo of order 0 can be deduced from the formula Π_θ,φ(P)=i2π(_θ P - _φ P) proved in an earlier work (coauthored with Gaarde). In the analysis of _θ P one need only modify p_m(x,ξ) in a neighborhood of e^iθR_+ this is known to be possible from Seeley’s 1967 work on complex powers.