Abstract:
We obtain some necessary and sufficient conditions for the strong continuity of representations of topological groups in reflexive Fréchet spaces. In particular, we show that a representation $\pi$ of a topological group $G$ in a reflexive Fréchet space is continuous in the strong operator topology if and only if for some number $q$, $0\le q<1$, and some neighbourhood $V$ of the identity element $e\in G$, for any neighbourhood $U$ of the zero element in $E$, its polar $\overset\circ{U}$ in the dual space $E^*$, any vector $\xi$ in $U$ and any element $f\in\overset\circ{U}$ the inequality $|f(\pi(g)\xi-\xi)|\le q$ holds for all $g\in V$.
Bibliography: 26 titles.
Keywords:locally convex space, polar, reflexive Fréchet space, topological group, continuity in the strong operator topology.
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