A Hille-Yosida type theorem in ordered convex cones

It is proved that the Laplace transform establishes a bijection between a class of resolvents (V-alpha)(alpha>0) and a class of semi-groups Phi of kernels, acting on an abstract ordered convex cone. The compactness ( in some weak topology) of the closed convex envelopes of the trajectories: Phi(t, x), t > 0, resp. of (nV(n))(k)x, n, k is an element of N, plays a central role in these results.