Garding domains for unitary representations of countable inductive limits of locally compact groups

Анотація

Let $G$ be the inductive limit of an increasing sequence of locally compact second countable groups $G_1\subset G_2 \subset \ldots$.  Given a strongly continuous unitary representation $U$ of $G$ in a separable Hilbert space $\mathcal{H}$, we construct an $U\text{-}$invariant, separable, nuclear, Montel (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.