Resumen de The sectorial projection defined from logarithms

Gerd Grubb

• For a classical elliptic pseudodifferential operator P of order >0 on a closed manifold X, such that the eigenvalues of the principal symbol pm(x,ξ) have arguments in ]θ,φ[ and ]φ,θ+2π[ (θ<φ<θ+2π), the sectorial projection Πθ,φ(P) is defined essentially as the integral of the resolvent along eiφR¯¯¯¯+∪eiθR¯¯¯¯+. In a recent paper, Booss-Bavnbek, Chen, Lesch and Zhu have pointed out that there is a flaw in several published proofs that Πθ,φ(P) is a ψdo of order 0; namely that pm(x,ξ) cannot in general be modified to allow integration of (pm(x,ξ)−λ)−1 along eiφR¯¯¯¯+∪eiθR¯¯¯¯+ simultaneously for all ξ. We show that the structure of Πθ,φ(P) as a ψdo of order 0 can be deduced from the formula Πθ,φ(P)=i2π(logθP−logφP) proved in an earlier work (coauthored with Gaarde). In the analysis of logθP one need only modify pm(x,ξ) in a neighborhood of eiθR¯¯¯¯+ this is known to be possible from Seeley's 1967 work on complex powers.