Quasinormable C0-Groups and Translation-Invariant Frechet Spaces of Type DE

Let E be a locally convex Hausdorff space satisfying the convex compact property and let (Tx)xRd be a locally equicontinuous C0-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 (Tx)xRd is also quasinormable. In particular, we obtain that the space of smooth vectors associated to a C0-group on a Banach space is always quasinormable. As an application, we show that the translation-invariant Frechet spaces of smooth functions of type DE (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 DE 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.