Quasinormable \(C_0\)-groups and translation-invariant Fréchet spaces of type \(\mathcal{D}_E\) (Q2323072)

The author first studies quasinormability in the context of \(C_0\)-groups. Namely, let \(E\) be a locally convex Hausdorff space satisfying the convex compact property and let \((T_x)_{\in\mathbb{R}^d}\) be a locally equicontinuous \(C_0\)-group of continuous linear operators on \(E\). The main result asserts that, if \(E\) is quasinormable, then the space of smooth vectors in \(E\) associated to \((T_x)_{\in\mathbb{R}^d}\), endowed with its natural locally convex topology, is also quasinormable. Since every Banach space satisfies the convex compact property and is quasinormable and every \(C_0\)-group on a Banach space is locally equicontinuous, it follows that the space of smooth vectors associated to a \(C_0\)-group on a Banach space is always quasinormable. The second part of this article is devoted to the study of the linear topological properties of the translation-invariant Fréchet spaces of smooth functions of type \({\mathcal D}_E\). In particular, the author shows that the space \({\mathcal D}_E\) is quasinormable for any translation-invariant Banach spaces of distributions \(E\).