关于光滑向量空间的一个不变性
On an invariance property of the space of smooth vectors
摘要：
设$(π, \mathcal H)$是（无限维）李群$G$的连续酉表示，并且$γ\: \mathbb R \to \mathrm{Aut}(G)$定义$\mathbb R$在$G$上的连续作用。设$π^\#(g,t) = π(g) U_t$定义了半直积群$G \rtimes_γ\mathbb R$的连续酉表示.本说明的第一个主要定理提供了$π$的光滑向量的空间$\mathcal H^\infty$在分别用于$f \in L^1(\mathbb R)$的算子$U_f = \int_\mathbb R f(t)U_t\, dt$下的不变性的准则，$f \in \mathcal S(\mathbb R)$。利用这一定理，我们证明了，对于复化李代数$\mathfrak g_{\mathbb C}$中适当定义的谱子空间$\mathfrak g_{\mathbb C}(E)$，$E \subseteq \mathbb R$，以及对于$\mathcal H^\infty$中适当定义的$U$，我们有\[ \mathsf{d}π(\mathfrak g_{\mathbb C}(E)) \mathcal H^\infty(F) \subseteq \mathcal H^\infty(E + F).\]
英文摘要：
Let $(π, \mathcal H)$ be a continuous unitary representation of the (infinite dimensional) Lie group $G$ and $γ\: \mathbb R \to \mathrm{Aut}(G)$ define a continuous action of $\mathbb R$ on $G$. Suppose that $π^\#(g,t) = π(g) U_t$ defines a continuous unitary representation of the semidirect product group $G \rtimes_γ\mathbb R$. The first main theorem of the present note provides criteria for the invariance of the space $\mathcal H^\infty$ of smooth vectors of $π$ 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 \[ \mathsf{d}π(\mathfrak g_{\mathbb C}(E)) \mathcal H^\infty(F) \subseteq \mathcal H^\infty(E + F).\]
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