Decompositions of Differentiable Semigroups

Abstract

A differentiable semigroup is a topological semigroup $(S, * )$ in which $S$ 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 ${C^1}$ retraction $\Phi$ of $U$ into the set of idempotents of $S$ so that $\Phi (x)\Phi (y) = \Phi (x)$ for $x$ and $y$ in $U$ and $x\Phi (x)$ is in the maximal subgroup of $S$ determined by $\Phi (x)$ 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 $\Phi$. 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 $\Phi$ implies that operators near an $e$ have nontrivial invariant subspaces. A dual right handed result holds.