Results in Mathematics

, 74:135 | Cite as

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

  • Andreas DebrouwereEmail author


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.


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

Mathematics Subject Classification

46E10 47D03 46A10 



  1. 1.
    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)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bastin, F., Ernst, B.: A criterion for \(CV(X)\) to be quasinormable. Results Math. 14, 223–230 (1988)MathSciNetCrossRefGoogle Scholar
  3. 3.
    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)Google Scholar
  4. 4.
    Bierstedt, K.D., Meise, R., Summers, W.H.: A projective description of weighted inductive limits. Trans. Am. Math. Soc. 272, 107–160 (1982)MathSciNetCrossRefGoogle Scholar
  5. 5.
    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)Google Scholar
  6. 6.
    Bonet, J., Dierolf, S.: On the lifting of bounded sets. Proc. Edinb. Math. Soc. 36, 277–281 (1993)CrossRefGoogle Scholar
  7. 7.
    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)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dimovski, P., Pilipović, S., Vindas, J.: New distribution spaces associated to translation-invariant Banach spaces. Monatsh. Math. 177, 495–515 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dimovski, P., Pilipović, S., Prangoski, B., Vindas, J.: Translation-modulation invariant Banach spaces of ultradistributions. J. Fourier Anal. Appl., in press (2018). Scholar
  10. 10.
    Feichtinger, H.G., Gröchenig, K.: Banach spaces related to integrable group representations and their atomic decompositions. I. J. Funct. Anal. 86, 307–340 (1989)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Grothendieck, A.: Sur les espaces \((F)\) et \((DF)\). Summa Brasil Math. 3, 57–123 (1954)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Hasumi, M.: Note on the \(n\)-dimensional tempered ultra-distributions. Tôhoku Math. J. 13, 94–104 (1961)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hille, E., Phillips, R.S.: Functional Analysis and Semi-groups. American Mathematical society, Providence (1957)zbMATHGoogle Scholar
  14. 14.
    Komura, T.: Semigroups of operators in locally convex spaces. J. Funct. Anal. 2, 258–296 (1968)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kuczma, M.: An Introduction to the Theory of Functional Equations and Inequalities, 2nd edn. Birkhäuser, Basel (2008)Google Scholar
  16. 16.
    Meise, R., Vogt, D.: A characterization of quasinormable Fréchet spaces. Math. Nachr. 122, 141–150 (1985)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Meise, R., Vogt, D.: Introduction to Functional Analysis. Clarendon Press, Oxford (1997)zbMATHGoogle Scholar
  18. 18.
    Qiu, J.H.: Local completeness and dual local quasi-completeness. Proc. Am. Math. Soc. 129, 1419–1425 (2001)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Rudin, W.: Functional Analsyis, 2nd edn. McGraw-Hill, New York (1991)Google Scholar
  20. 20.
    Schwartz, L.: Espaces de fonctions différentiables à valeurs vectorielles. J. d’Analyse Math. 4, 88–148 (1954)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Schwartz, L.: Théorie des Distributions. Hermann, Paris (1966)zbMATHGoogle Scholar
  22. 22.
    Valdivia, M.: On quasinormable echelon spaces. Proc. Edinb. Math. Soc. 24, 73–80 (1981)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Valdivia, M.: Basic sequences in the dual of a Fréchet space. Math. Nachr. 231, 169–185 (2001)MathSciNetCrossRefGoogle Scholar
  24. 24.
    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)Google Scholar
  25. 25.
    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)Google Scholar
  26. 26.
    Vogt, D.: On the functors \(\text{ Ext }^1(E;F)\) for Fréchet spaces. Studia Math. 85, 163–197 (1987)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Wolf, E.: Weighted Fréchet spaces of holomorphic functions. Studia Math. 174, 255–275 (2006)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Wolf, E.: A note on quasinormable weighted Fréchet spaces of holomorphic functions. Bull. Belg. Math. Soc. Simon Stevin 14, 587–593 (2007)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Wolf, E.: Quasinormable weighted Fréchet spaces of entire functions. Bull. Belg. Math. Soc. Simon Stevin 16, 351–360 (2009)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

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

Personalised recommendations