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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 thesemidirect product group $G \rtimes_\gamma \mathbb R$. The first main theoremof the present note provides criteria for the invariance of the space $\mathcalH^\infty$ of smooth vectors of $\pi$ under the operators $U_f = \int_\mathbb Rf(t)U_t\, dt$ 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 Liealgebra $\mathfrak g_{\mathbb C}$, and $\mathcal H^\infty(F)$, $F\subseteq\mathbb R$, for $U$ in $\mathcal H^\infty$, we have \[ \mathsf{d}\pi(\mathfrakg_{\mathbb C}(E)) \mathcal H^\infty(F) \subseteq \mathcal H^\infty(E + F).\]

On an invariance property of the space of smooth vectors