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

## TL;DR

This paper demonstrates that the space of smooth vectors for certain $C_0$-groups on locally convex spaces is quasinormable, and applies this to show that specific translation-invariant Fréchet spaces are quasinormable, answering open questions.

## Contribution

It proves quasinormability of smooth vector spaces for $C_0$-groups on locally convex spaces and applies this to translation-invariant Fréchet spaces, resolving previously posed questions.

## Key findings

- The space of smooth vectors is quasinormable if $E$ is quasinormable.
- Translation-invariant Fréchet spaces of type $\\mathcal{D}_E$ are quasinormable.
- $\\mathcal{D}_E$ is not Montel if $E$ is a solid translation-invariant Banach space.

## 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 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.