  We study the minimal and maximal closed extension of a differential operator
A on a manifold B with conical singularities, when A acts as an unbounded
operator on weighted L^p-spaces over B, 1 < p < \infty. Under suitable
ellipticity assumptions we can define a family of complex powers A^z. We also
obtain sufficient information on the resolvent of A to show the boundedness of
the purely imaginary powers. Examples concern unique solvability and maximal
regularity for the solution of the Cauchy problem for the Laplacian on conical
manifolds as well as certain quasilinear diffusion equations.
