This article is cited in 3 papers

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

A. I. Danilenko

Department of Mechanics and Mathematics, Kharkov State University, 4, Svobody Sq., 310077, Kharkov, Ukraine

Abstract: 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.

Received: 09.10.1995

Language: English