On an invariance property of the space of smooth vectors

Abstract

Let $(\pi, \mathcal H)$ be a continuous unitary representation of the (infinite dimensional) Lie group $G$ and $\gamma \: \mathbb R \to \mathrm{Aut}(G)$ define a continuous action of $\mathbb R$ on $G$. Suppose that $\pi^\#(g,t) = \pi(g) U_t$ defines a continuous unitary representation of the semidirect product group $G \rtimes_\gamma \mathbb R$. 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 $f \in L^1(\mathbb R)$, resp., $f \in \mathcal S(\mathbb R)$. Using this theorem we 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 $U$ in $\mathcal H^\infty$, we have \[ \mathsf{d}\pi(\mathfrak g_{\mathbb C}(E)) \mathcal H^\infty(F) \subseteq \mathcal H^\infty(E + F).\]