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 groupsin Frechet spaces that are dual to some locally convex spaces by operators adjoint to continuous linear operators in a predual spaceIn particular, it is shownthat a representation $\pi$ of a topological group $G$ on a Frechet space $E$ dual to a locally convex space $E_*$ by adjoint operators is continuous inthe 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 anyneighbourhood $U$ 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 $g\in V$.