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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Decompositions of differentiable semigroups

Author: John P. Holmes
Journal: Proc. Amer. Math. Soc. 110 (1990), 1111-1118
MSC: Primary 22A15; Secondary 47A15, 47H20, 58B25
MathSciNet review: 1031665
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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.

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Keywords: $ {C^1}$ semigroup, idempotent, retraction, invariant subspace
Article copyright: © Copyright 1990 American Mathematical Society