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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Decompositions of differentiable semigroups
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by John P. Holmes PDF
Proc. Amer. Math. Soc. 110 (1990), 1111-1118 Request permission

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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Additional Information
  • © Copyright 1990 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 110 (1990), 1111-1118
  • MSC: Primary 22A15; Secondary 47A15, 47H20, 58B25
  • DOI: https://doi.org/10.1090/S0002-9939-1990-1031665-5
  • MathSciNet review: 1031665