Quasinormable $C_0$-groups and translation-invariant Fr\'echet spaces of type ${\mathcal{D}}_E$

Let $E$ be a locally convex Hausdorff space satisfying the convex compact property and let $(T_x{)}_{x\in {\mathbb{R}}^d}$ be a locally equicontinuous $C_0$-group of linear continuous operators on $E$. In this article, we show that if $E$ is quasinormable, then the space of smooth vectors in $E$ associated to $(T_x{)}_{x\in {\mathbb{R}}^d}$ is also quasinormable. In particular, we obtain that the space of smooth vectors associated to a $C_0$-group on a Banach space is always quasinormable. As an application, we show that the translation-invariant Fr'echet spaces of smooth functions of type ${\mathcal{D}}_E$ [8] are quasinormable, thereby settling the question posed in [8, Remark 7]. Furthermore, we show that ${\mathcal{D}}_E$ is not Montel if $E$ is a solid translation-invariant Banach space of distributions [10]. This answers the question posed in [8, Remark 6] for the class of solid translation-invariant Banach spaces of distributions.