Differentiable vectors and unitary representations of Fréchet–Lie supergroups

TLDR: In this article, the Trotter property of locally convex Lie groups has been studied in the context of automorphisms of principal bundles over compact smooth manifolds, and it has been shown that for smooth (resp., analytic) unitary representations of Frechet-Lie supergroups, the common domain of the k-fold products of the operators can be extended to the space of analytic vectors for G.

Abstract: A locally convex Lie group G has the Trotter property if, for every \(x_1, x_2 \in \mathfrak{g }\), $$\begin{aligned} \exp _G(t(x_1 + x_2))=\lim _{n \rightarrow \infty } \left(\exp _G\left(\frac{t}{n}x_1\right)\exp _G\left(\frac{t}{n}x_2\right)\right)^n \end{aligned}$$ holds uniformly on compact subsets of \(\mathbb{R }\). All locally exponential Lie groups have this property, but also groups of automorphisms of principal bundles over compact smooth manifolds. A key result of the present article is that, if G has the Trotter property, \(\pi : G \rightarrow {\mathrm{GL}}(V)\) is a continuous representation of G on a locally convex space, and \(v \in V\) is a vector such that \(\overline{\mathtt{d}\pi }(x)v :=\frac{d}{dt}|_{t=0} \pi (\exp _G(tx))v\) exists for every \(x \in \mathfrak{g }\), then the map \(\mathfrak{g }\rightarrow V,x \mapsto \overline{\mathtt{d}\pi }(x)v\) is linear. Using this result we conclude that, for a representation of a locally exponential Frechet–Lie group G on a metrizable locally convex space, the space of \(\mathcal{C }^{k}\)-vectors coincides with the common domain of the k-fold products of the operators \(\overline{\mathtt{d}\pi }(x)\). For unitary representations on Hilbert spaces, the assumption of local exponentiality can be weakened to the Trotter property. As an application, we show that for smooth (resp., analytic) unitary representations of Frechet–Lie supergroups \((G,\mathfrak{g })\) where G has the Trotter property, the common domain of the operators of \(\mathfrak{g }=\mathfrak{g }_{\overline{0}}\oplus \mathfrak{g }_{\overline{1}}\) can always be extended to the space of smooth (resp., analytic) vectors for G.