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, 74:135 | Cite as

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

  • Andreas DebrouwereEmail author
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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 Fréchet 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 Gröchenig 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.

Keywords

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

Mathematics Subject Classification

46E10 47D03 46A10 
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Notes

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Authors and Affiliations

  1. 1.Department of Mathematics: Analysis, Logic and Discrete MathematicsGhent UniversityGentBelgium

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