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 Fre<acute accent>chet spaces. In particular, we show that a representation pi of a topological group G in a reflexive Fre<acute accent>chet space is continuous in the strong operator topology if and only if for some number q, 0 < q < 1, and some neighbourhood V of the identity element e is an element of G, for any neighbourhood U-degrees of the zero element in E, its polar U in the dual space E-& lowast;, any vector xi(degrees) in U and any element f is an element of U-degrees the inequality f (pi(g)xi-xi) <= q holds for all g is an element of V. Bibliography: 26 titles.