Ellipticity and invertibility in the cone algebra onL p -Sobolev spaces

Given a manifoldB with conical singularities, we consider the cone algebra with discrete asymptotics, introduced by Schulze, on a suitable scale ofL _p -Sobolev spaces. Ellipticity is proven to be equivalent to the Fredholm property in these spaces; it turns out to be independent of the choice ofp. We then show that the cone algebra is closed under inversion: whenever an operator is invertible between the associated Sobolev spaces, its inverse belongs to the calculus. We use these results to analyze the behaviour of these operators onL _p (B).