The sectorial projection defined from logarithms

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,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^{iϕ}R_+}\cup {e^{iθ}R_+}. In a recent paper, Booss-Bavnbek, Chen, Lesch and Zhu have pointed out that there is a flaw in several published proofs that ¶_{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^{iϕ}R_+}\cup {e^{iθ}R_+} 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)= (i/(2π))(\log_θ(P) - \log_ϕ(P)) proved in an earlier work (coauthored with Gaarde). In the analysis of \log_θ(P) one need only modify p_m(x,ξ) in a neighborhood of e^{iθ}R_+; this is known to be possible from Seeley's 1967 work on complex powers.

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