A criterion for the weak continuityof representations of topological groups in dual Frechet spaces

Sufficient conditions are obtained for the weak continuity of representations of topological groups in Fréchet spaces that are dual to some locally convex spaces by operators adjoint to continuous linear operators in a predual space In particular, it is shown that a representation $\pi$ of a topological group $G$ on a Fréchet space $E$ dual to a locally convex space $E_*$ by adjoint operators is continuous in the weak$^*$ operator topology if, for some number $q$, $0\le q<1$, there is a neighbourhood $V$ of the neutral element $e$ of $G$ such that, for any neighbourhood $U$ of the zero element in $E$, for its polar $\mathring{U}$ in $E^*$, and for any vector $\xi$ in $U$ and any element $\varphi\in\mathring{U}$ the inequality $|(\pi(g)\xi-\xi)(\varphi)|\le q$ holds for each $g\in V$.