Closed universal subspaces of spaces of infinitely differentiable functions
Annales de l'Institut Fourier, Volume 64 (2014) no. 1, pp. 297-325.

We exhibit the first examples of Fréchet spaces which contain a closed infinite dimensional subspace of universal series, but no restricted universal series. We consider classical Fréchet spaces of infinitely differentiable functions which do not admit a continuous norm. Furthermore, this leads us to establish some more general results for sequences of operators acting on Fréchet spaces with or without a continuous norm. Additionally, we give a characterization of the existence of a closed subspace of universal series in the Fréchet space 𝕂 .

On exhibe les premiers exemples d’espaces de Fréchet contenant un sous-espace fermé de dimension infinie de séries universelles, mais ne contenant aucune série universelle restreinte. Pour cela, on considère les espaces de Fréchet classiques de fonctions indéfiniment dérivables qui n’admettent pas de norme continue. On établit alors des résultats plus généraux pour des suites d’opérateurs qui agissent sur des espaces de Fréchet avec ou sans norme continue. Enfin, on caractérise complètement l’existence de sous-espaces fermés de séries universelles dans l’espace de Fréchet 𝕂 .

DOI: 10.5802/aif.2848
Classification: 30K05, 41A58, 26E10, 46E15, 47A16
Keywords: infinitely differentiable real functions, spaceability, universality, universal series, Taylor series
Mot clés : fonctions indéfiniment dérivables, sous-espaces fermés universels, universalité, séries universelles, séries de Taylor.
Charpentier, Stéphane 1; Menet, Quentin 2; Mouze, Augustin 3

1 Laboratoire Paul Painlevé, UMR 8524, Université Lille 1, Cité Scientifique, 59650 Villeneuve d’Ascq
2 Institut de Mathématique, Université de Mons, 20 Place du Parc, 7000 Mons, Belgique
3 Laboratoire Paul Painlevé, UMR 8524, Current address: École Centrale de Lille, Cité Scientifique, BP48, 59651 Villeneuve d’Ascq cedex
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Charpentier, Stéphane; Menet, Quentin; Mouze, Augustin. Closed universal subspaces of spaces of infinitely differentiable functions. Annales de l'Institut Fourier, Volume 64 (2014) no. 1, pp. 297-325. doi : 10.5802/aif.2848. http://archive.numdam.org/articles/10.5802/aif.2848/

[1] Aron, R. Linearity in non-linear situations, Advanced courses of mathematical analysis. II, World Sci. Publ., Hackensack, NJ, 2007, pp. 1-15 | MR | Zbl

[2] Bayart, F. Linearity of sets of strange functions, Michigan Math. J., Volume 53 (2005) no. 2, pp. 291-303 | DOI | MR | Zbl

[3] Bayart, F.; Grosse-Erdmann, K.-G.; Nestoridis, V.; Papadimitropoulos, C. Abstract theory of universal series and applications, Proc. London Math. Soc., Volume 96 (2008), pp. 417-463 | DOI | MR | Zbl

[4] Bonet, J. A problem on the structure of Fréchet spaces, Rev. R. Acad. Cien. Serie A. Mat., Volume 104 (2010), pp. 427-434 | DOI | MR | Zbl

[5] Bonet, J.; Martìnez-Giménez, F.; Peris, A. Universal and chaotic multipliers on spaces of operators, J. Math. Anal. Appl., Volume 294 (2004), pp. 599-611 | DOI | MR | Zbl

[6] Borel, E. Sur quelques points de la théorie des fonctions, Ann. Sci. École Norm. Sup., Volume 12 (1895), pp. 9-55 | MR

[7] Bés, J.; Conejero, J. A. Hypercyclic subspaces in omega, J. Math. Anal. Appl., Volume 316 (2006), pp. 16-23 | DOI | MR | Zbl

[8] Charpentier, S. On the closed subspaces of universal series in Banach spaces and Fréchet spaces, Studia Math., Volume 198 (2010), pp. 121-145 | DOI | MR | Zbl

[9] Grosse Erdmann, K-G. Universal families and hypercyclic operators, Bull. Amer. Math. Soc. (N.S.), Volume 36 (1999), pp. 345-381 | DOI | MR | Zbl

[10] Komatsu, H. Ultradistributions, I. Structure theorems and a characterization, J. Fac. Sci. Univ. Tokyo, Sect. 1A, Volume 20 (1973), pp. 25-105 | MR | Zbl

[11] León Saavedra, F.; Müller, V. Hypercyclic sequences of operators, Studia Math., Volume 175 (2006), pp. 1-18 | DOI | MR | Zbl

[12] Menet, Q. Sous-espace fermés de séries universelles sur un espace de Fréchet, Studia Math., Volume 207 (2011), pp. 181-195 | DOI | MR | Zbl

[13] Menet, Q. Hypercyclic subspaces and weighted shifts, 2012 (preprint)

[14] Mouze, A.; Nestoridis, V. Universality and ultradifferentiable functions: Fekete’s Theorem, Proc. Amer. Math. Soc., Volume 138 (2010) no. 11, pp. 3945-3955 | DOI | MR | Zbl

[15] Petersson, H. Hypercyclic subspaces for Fréchet space operators, J. Math. Anal. Appl., Volume 319 (2006), pp. 764-782 | DOI | MR | Zbl

[16] Petzsche, H-J. On E. Borel’s theorem, Math. Ann., Volume 282 (1988), pp. 299-313 | DOI | Zbl

[17] Pál, G. Zwei kleine Bemerkungen, Tohoku Math. J., Volume 6 (1914/15), pp. 42-43

[18] Seleznev, A. I. On universal power series, Math. Sbornik N.S., Volume 28 (1951), pp. 453-460 | MR | Zbl

[19] Tsirivas, N. Simultaneous approximation by universal series, Math. Nachr., Volume 283 (2010), pp. 909-920 | DOI | MR | Zbl

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