Difference between revisions of "Invariant subspace of a representation"
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| − | A vector (respectively, a closed vector) subspace | + | {{TEX|auto}} |
| + | {{TEX|done}} | ||
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| + | '' $ \pi $ | ||
| + | of a group (algebra, ring, semi-group) $ X $ | ||
| + | in a vector space (or topological vector space) $ E $'' | ||
| + | |||
| + | A vector (respectively, a closed vector) subspace $ F \subset E $ | ||
| + | such that for any $ \xi \in F $ | ||
| + | and any $ x \in X $ | ||
| + | one has $ \pi ( x ) \xi \in F $. | ||
| + | If $ P $ | ||
| + | is a projection operator from $ E $ | ||
| + | onto $ F $, | ||
| + | then $ F $ | ||
| + | is an invariant subspace of $ \pi $ | ||
| + | if and only if $ P \pi ( x ) P = \pi ( x ) P $ | ||
| + | for all $ x \in X $. | ||
| + | The subspace $ \{ 0 \} $ | ||
| + | in $ E $ | ||
| + | is invariant for any representation in $ E $; | ||
| + | it is called the trivial invariant subspace; the remaining invariant subspaces (if there are any) are called non-trivial. See also [[Contraction of a representation|Contraction of a representation]]; [[Completely-reducible set|Completely-reducible set]]; [[Irreducible representation|Irreducible representation]]. | ||
Latest revision as of 22:13, 5 June 2020
$ \pi $
of a group (algebra, ring, semi-group) $ X $
in a vector space (or topological vector space) $ E $
A vector (respectively, a closed vector) subspace $ F \subset E $ such that for any $ \xi \in F $ and any $ x \in X $ one has $ \pi ( x ) \xi \in F $. If $ P $ is a projection operator from $ E $ onto $ F $, then $ F $ is an invariant subspace of $ \pi $ if and only if $ P \pi ( x ) P = \pi ( x ) P $ for all $ x \in X $. The subspace $ \{ 0 \} $ in $ E $ is invariant for any representation in $ E $; it is called the trivial invariant subspace; the remaining invariant subspaces (if there are any) are called non-trivial. See also Contraction of a representation; Completely-reducible set; Irreducible representation.
How to Cite This Entry:
Invariant subspace of a representation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Invariant_subspace_of_a_representation&oldid=47419
Invariant subspace of a representation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Invariant_subspace_of_a_representation&oldid=47419
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article