On an invariance property of the space of smooth vectors

Let $(\pi ,\mathcal{H})$ be a continuous unitary representation of the (infinite dimensional) Lie group $G$ and $\gamma : \mathbb R \to $\mathrm{A}\mathrm{u}\mathrm{t}$(G)$defineacontinuousactionof$\mathbb R$on$G$\mathrm{.}Supposethat$\pi^#(g,t) = \pi(g) U_t$ defines a continuous unitary representation of the semidirect product group $G{⋊}_{\gamma }\mathbb{R}$. The first main theorem of the present note provides criteria for the invariance of the space $\mathcal H^\infty$ofsmoothvectorsof$\pi$undertheoperators$U_f = \int_\mathbb R f(t)U_t, dt$for$f \in L^1(\mathbb R)$,resp\mathrm{.},$f \in \mathcal S(\mathbb R)$. 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 ${\mathcal{H}}^{\mathrm{\infty }}(F)$, $F\subseteq \mathbb R$,for$U$in$\mathcal H^\infty$, we have $$\mathsf{d}\pi ({\mathfrak{g}}_{\mathbb{C}}(E)){\mathcal{H}}^{\mathrm{\infty }}(F)\subseteq {\mathcal{H}}^{\mathrm{\infty }}(E+F)\mathrm{.}$$