Decompositions of differentiable semigroups

Abstract

A differentiable semigroup is a topological semigroup ( S , ∗ ) (S, * ) in which S S is a differentiable manifold based on a Banach space and the associative multiplication function * is continuously differentiable. If e e is an idempotent element of such a semigroup we show that there is an open set U U containing e e so that there is a C 1 {C^1} retraction Φ \Phi of U U into the set of idempotents of S S so that Φ ( x ) Φ ( y ) = Φ ( x ) \Phi (x)\Phi (y) = \Phi (x) for x x and y y in U U and x Φ ( x ) x\Phi (x) is in the maximal subgroup of S S determined by Φ ( x ) \Phi (x) for each x x in U U . This leads to a natural decomposition of S S near e e into the union of a collection of mutually disjoint and mutually homeomorphic local differentiable subsemigroups whose intersections with U U are the point inverses under Φ \Phi . In case S S is the semigroup under composition of continuous linear transformations on a Banach space, in the case of a nontrivial idempotent e e , the existence of Φ \Phi implies that operators near an e e have nontrivial invariant subspaces. A dual right handed result holds.