TEMPERED OPERATORS AND THE HEAT KERNEL AND COMPLEX POWERS OF ELLIPTIC PSEUDODIFFERENTIAL OPERATORS

The resolvent (A – λ)−1 of an elliptic b-pseudodifferential operator on a compact manifold with corners (of arbitrary codimension) is shown to lie in a calculus of operators, tempered in the parameter λ in a special way. We show that the Laplace and Mellin transforms, with respect to λ, of these tempered operators can be defined and that they have Schwartz kernels which can be described geometrically. As a corollary, we obtain the structures of the kernels of the heat operator and complex powers of b-pseudodifferential operators, as the heat operator and complex powers are the Laplace and Mellin transforms, respectively, of the resolvent. The heat operator and complex powers are then used to generalize the index formula of Atiyah, Patodi, and Singer for Dirac operators on manifolds with boundary to Fredholm b-pseudodifferential operators on arbitrary compact manifolds with corners.