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.
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References
Albanese, A., Bonet, J., Ricker, W.: \(C_0\)-semigroups and mean ergodic operators in a class of Fréchet spaces. J. Math. Anal. Appl. 365, 142–157 (2010)
Bastin, F., Ernst, B.: A criterion for \(CV(X)\) to be quasinormable. Results Math. 14, 223–230 (1988)
Bierstedt, K.D., Meise, R.: Distinguished echelon spaces and the projective description of weighted inductive limits of type \(V_dC(X)\), pp. 169–226, In: Aspects of Mathematics and its Applications, North-Holland Math. Library 34, Amsterdam, (1986)
Bierstedt, K.D., Meise, R., Summers, W.H.: A projective description of weighted inductive limits. Trans. Am. Math. Soc. 272, 107–160 (1982)
Bierstedt, K.D., Meise, R., Summers, W.H.: Köthe sets and Köthe sequence spaces, pp. 27–91, In: Functional Analysis, Holomorphy and Approximation Theory, North-Holland Math. Stud. 71, Amsterdam (1982)
Bonet, J., Dierolf, S.: On the lifting of bounded sets. Proc. Edinb. Math. Soc. 36, 277–281 (1993)
Dierolf, P., Dierolf, S.: Topological properties of the dual pair \(\langle \dot{\cal{B}}(\Omega )^{\prime }, \dot{\cal{B}}(\Omega )^{\prime \prime } \rangle \). Pac. J. Math. 108, 51–82 (1983)
Dimovski, P., Pilipović, S., Vindas, J.: New distribution spaces associated to translation-invariant Banach spaces. Monatsh. Math. 177, 495–515 (2015)
Dimovski, P., Pilipović, S., Prangoski, B., Vindas, J.: Translation-modulation invariant Banach spaces of ultradistributions. J. Fourier Anal. Appl., in press (2018). https://doi.org/10.1007/s00041-018-9610-x.
Feichtinger, H.G., Gröchenig, K.: Banach spaces related to integrable group representations and their atomic decompositions. I. J. Funct. Anal. 86, 307–340 (1989)
Grothendieck, A.: Sur les espaces \((F)\) et \((DF)\). Summa Brasil Math. 3, 57–123 (1954)
Hasumi, M.: Note on the \(n\)-dimensional tempered ultra-distributions. Tôhoku Math. J. 13, 94–104 (1961)
Hille, E., Phillips, R.S.: Functional Analysis and Semi-groups. American Mathematical society, Providence (1957)
Komura, T.: Semigroups of operators in locally convex spaces. J. Funct. Anal. 2, 258–296 (1968)
Kuczma, M.: An Introduction to the Theory of Functional Equations and Inequalities, 2nd edn. Birkhäuser, Basel (2008)
Meise, R., Vogt, D.: A characterization of quasinormable Fréchet spaces. Math. Nachr. 122, 141–150 (1985)
Meise, R., Vogt, D.: Introduction to Functional Analysis. Clarendon Press, Oxford (1997)
Qiu, J.H.: Local completeness and dual local quasi-completeness. Proc. Am. Math. Soc. 129, 1419–1425 (2001)
Rudin, W.: Functional Analsyis, 2nd edn. McGraw-Hill, New York (1991)
Schwartz, L.: Espaces de fonctions différentiables à valeurs vectorielles. J. d’Analyse Math. 4, 88–148 (1954)
Schwartz, L.: Théorie des Distributions. Hermann, Paris (1966)
Valdivia, M.: On quasinormable echelon spaces. Proc. Edinb. Math. Soc. 24, 73–80 (1981)
Valdivia, M.: Basic sequences in the dual of a Fréchet space. Math. Nachr. 231, 169–185 (2001)
Vogt, D.: Sequence space representations of spaces of test functions and distributions, pp. 405–443, In: Functional analysis, Holomorphy, and Approximation Theory, Lecture Notes in Pure and Appl. Math. 83, New York (1983)
Vogt, D.: Some results on continuous linear maps between Fréchet spaces, pp. 349–381, In: Functional analysis: Surveys and Recent Results III, North-Holland Math. Stud. 90, Amsterdam (1984)
Vogt, D.: On the functors \(\text{ Ext }^1(E;F)\) for Fréchet spaces. Studia Math. 85, 163–197 (1987)
Wolf, E.: Weighted Fréchet spaces of holomorphic functions. Studia Math. 174, 255–275 (2006)
Wolf, E.: A note on quasinormable weighted Fréchet spaces of holomorphic functions. Bull. Belg. Math. Soc. Simon Stevin 14, 587–593 (2007)
Wolf, E.: Quasinormable weighted Fréchet spaces of entire functions. Bull. Belg. Math. Soc. Simon Stevin 16, 351–360 (2009)
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The author is supported by FWO-Vlaanderen, via the postdoctoral grant 12T0519N.
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Debrouwere, A. Quasinormable \(\varvec{C_0}\)-Groups and Translation-Invariant Fréchet Spaces of Type \(\varvec{\mathcal {D}_E}\). Results Math 74, 135 (2019). https://doi.org/10.1007/s00025-019-1055-2
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DOI: https://doi.org/10.1007/s00025-019-1055-2