Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

8
The categories of this paper are: hep-ex
0
The categories of this paper are: hep-ph
29
The categories of this paper are: funct-an
31
The categories of this paper are: ao-sci
28
The categories of this paper are: alg-geom
19
The categories of this paper are: eess
1
The categories of this paper are: cs
5
The categories of this paper are: gr-qc
16
The categories of this paper are: stat
37
The categories of this paper are: supr-con
4
The categories of this paper are: cond-mat
15
The categories of this paper are: nucl-ex
11
The categories of this paper are: hep-lat
10
The categories of this paper are: q-bio
30
The categories of this paper are: q-alg
20
The categories of this paper are: acc-phys