A~criterion for the strong continuity of~representations of~topological groups in reflexive Fr\'echet spaces

A. I. Shtern

Sbornik. Mathematics, Tome 216 (2025) no. 1, pp. 132-139

Résumé

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 $q$, $0\le q1$, 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.

Détail

Keywords: locally convex space, polar, reflexive Fréchet space, topological group, continuity in the strong operator topology.