## Kyoto Journal of Mathematics

### On an invariance property of the space of smooth vectors

#### Abstract

Let $(\pi,\mathcal{H})$ be a continuous unitary representation of the (infinite-dimen- sional) Lie group $G$, and let $\gamma\colon \mathbb{R}\to\operatorname{Aut}(G)$ be a group homomorphism which defines a continuous action of $\mathbb{R}$ on $G$ by Lie group automorphisms. Let $\pi^{\#}(g,t)=\pi(g)U_{t}$ be a continuous unitary representation of the semidirect product group $G\rtimes_{\gamma}\mathbb{R}$ on $\mathcal{H}$. The first main theorem of the present note provides criteria for the invariance of the space $\mathcal{H}^{\infty}$ of smooth vectors of $\pi$ under the operators $U_{f}=\int_{\mathbb{R}}f(t)U_{t}dt$ for $f\in L^{1}(\mathbb{R})$ and $f\in\mathcal{S}(\mathbb{R})$, respectively. When $\mathfrak{g}$ is complete and the actions of $\mathbb{R}$ on $G$ and $\mathfrak{g}$ are continuous, we use the above theorem to show that, for suitably defined spectral subspaces $\mathfrak{g}_{\mathbb{C}}(E)$, $E\subseteq\mathbb{R}$, in the complexified Lie algebra $\mathfrak{g}_{\mathbb{C}}$ and $\mathcal{H}^{\infty}(F)$, $F\subseteq\mathbb{R}$, for $U_{t}$ in $\mathcal{H}^{\infty}$, we have

$$\mathrm{d}\pi(\mathfrak{g}_{\mathbb{C}}(E))\mathcal{H}^{\infty}(F)\subseteq\mathcal{H}^{\infty}(E+F).$$

#### Article information

Source
Kyoto J. Math., Volume 55, Number 3 (2015), 501-515.

Dates
Revised: 24 April 2014
Accepted: 25 April 2014
First available in Project Euclid: 9 September 2015

https://projecteuclid.org/euclid.kjm/1441824031

Digital Object Identifier
doi:10.1215/21562261-3089019

Mathematical Reviews number (MathSciNet)
MR3395973

Zentralblatt MATH identifier
1325.22012

#### Citation

Neeb, Karl-Hermann; Salmasian, Hadi; Zellner, Christoph. On an invariance property of the space of smooth vectors. Kyoto J. Math. 55 (2015), no. 3, 501--515. doi:10.1215/21562261-3089019. https://projecteuclid.org/euclid.kjm/1441824031

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