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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.

References [Enhancements On Off] (What's this?)

  • [1] B. Beauzamy, Introduction to operator theory and invariant subspaces, North-Holland, Amsterdam, New York, Oxford, and Tokyo, 1988. MR 967989 (90d:47001)
  • [2] G. Birkhoff, Analytical groups, Trans. Amer. Math. Soc. 43 (1938), 61-101. MR 1501934
  • [3] J. Dieudonné, Foundations of modern analysis, vol. 1, Academic Press, New York and London, 1960. MR 0120319 (22:11074)
  • [4] N. Dunford and J. T. Schwartz, Linear operators Part I: the general theory, Interscience, New York and London, 1958. MR 1009162 (90g:47001a)
  • [5] J. P. Holmes, Differentiable projections and differentiable semigroups, Proc. Amer. Math. Soc. 41 (1972), 391-395. MR 0375378 (51:11572)
  • [6] -, Differentiable semigroups, Colloq. Math. 32 (1974), 99-104. MR 0369599 (51:5832)
  • [7] -, Rees products in differentiable semigroups, Semigroup Forum 25 (1982), 145-152. MR 663175 (83i:22003)
  • [8] -, A rank theorem for infinite dimensional spaces, Proc. Amer. Math. Soc. 50 (1975), 358-364. MR 0383452 (52:4333)

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

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