Quasinormable $$\varvec{C_0}$$-Groups and Translation-Invariant Fréchet Spaces of Type $$\varvec{\mathcal {D}_E}$$

Abstract

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 Frechet spaces of smooth functions of type $$\mathcal {D}_E$$ (Dimovski et al. in Monatsh Math 177:495–515, 2015) 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 (Feichtinger and Grochenig in J Funct Anal 86:307–340, 1989). This answers the question posed in [8, Remark 6] for the class of solid translation-invariant Banach spaces of distributions.