Decompositions of Differentiable Semigroups

A differentiable semigroup is a topological semigroup (5, *) in which 5 is a differentiable manifold based on a Banach space and the associative multiplication function * is continuously differentiable. If e is an idempotent element of such a semigroup we show that there is an open set U containing e so that there is a C retraction of U into the set of idempotents of S so that (x)<$(y) = <3>(x) for x and y in U and x(x) is in the maximal subgroup of 5 determined by O(x) for each x in U . This leads to a natural decomposition of S near e into the union of a collection of mutually disjoint and mutually homeomorphic local differentiable subsemigroups whose intersections with U are the point inverses under . In case S is the semigroup under composition of continuous linear transformations on a Banach space, in the case of a nontrivial idempotent e , the existence of implies that operators near an e have nontrivial invariant subspaces. A dual right handed result holds.