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

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Abstract

We obtain some necessary and sufficient conditions for the strong continuity of representations of topological groups in reflexive Frechet spaces. In particular, we show that a representation $\pi$ of a topological group $G$ in a reflexive Frechet 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.

About the authors

Alexander Isaakovich Shtern

Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia; Moscow Center of Fundamental and Applied Mathematics, Moscow, Russia; Scientific Research Institute for System Studies of the Russian Academy of Science, Moscow, Russia

Author for correspondence.
Email: rroww@mail.ru

Candidate of physico-mathematical sciences, Associate professor

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