A criterion for the strong continuity of representations of topological groups in reflexive Frechet spaces

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.