G{\aa}rding domains for unitary representations of countable inductive limits of locally compact groups

A. I. Danilenko

Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 3 (1996), pp. 231-260.

Résumé

Let $G$ be the inductive limit of an increasing sequence of locally compact second countable groups $G_1\subset G_2\subset\cdots$. Given a strongly continuous unitary representation $U$ of $G$ in a separable Hilbert space $\mathcal H$, we construct an $U$-invariant, separable, nuclear, Montel $(\mathrm{DF})$-space $\mathcal F$ which is densely (topologically) embedded in $\mathcal H$ and such that the restriction of $U$ to $\mathcal F$ is a weakly continuous representation of $G$ by continuous linear operators in $\mathcal F$. Moreover, $\mathcal F$ is a domain of essential self-adjointness for the generator of each one-parameter subgroup of $G$, and all such generators keep $\mathcal F$ invariant.