On the Index of Elliptic Operators on Manifolds with Boundary

An extension of the index theorem of Atiyah-Patodi-Singer for Dirac-type operators on manifolds with boundary to general elliptic b-pseudodifferential operators is given. First basic results about the complex powers of these operators are established. Then the main formula for the index of an elliptic b-pseudo-differential operator acting on an r-weighted Sobolev space, r ∈ R, is proved. This expresses the index as the sum of an interior contribution, given in terms of regularized zeta functions, and a boundary contribution generalizing the eta invariant of Atiyah-Patodi-Singer. This second term measures the asymmetry of the boundary spectrum of the operator, a discrete set in the complex plane, with respect to the line {z ∈ C; Im z = −r}.