Closed universal subspaces of spaces of infinitely differentiable functions  [ Sous-espaces fermés universels dans des espaces de fonctions indéfiniment dérivables ]
Annales de l'Institut Fourier, Tome 64 (2014) no. 1, p. 297-325
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 .
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 .
DOI : https://doi.org/10.5802/aif.2848
Classification:  30K05,  41A58,  26E10,  46E15,  47A16
Mots clés: fonctions indéfiniment dérivables, sous-espaces fermés universels, universalité, séries universelles, séries de Taylor.
@article{AIF_2014__64_1_297_0,
     author = {Charpentier, St\'ephane and Menet, Quentin and Mouze, Augustin},
     title = {Closed universal subspaces of spaces of~infinitely differentiable functions},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {64},
     number = {1},
     year = {2014},
     pages = {297-325},
     doi = {10.5802/aif.2848},
     mrnumber = {3330550},
     zbl = {06387275},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2014__64_1_297_0}
}
Charpentier, Stéphane; Menet, Quentin; Mouze, Augustin. Closed universal subspaces of spaces of infinitely differentiable functions. Annales de l'Institut Fourier, Tome 64 (2014) no. 1, pp. 297-325. doi : 10.5802/aif.2848. http://www.numdam.org/item/AIF_2014__64_1_297_0/

[1] Aron, R. Linearity in non-linear situations, Advanced courses of mathematical analysis. II, World Sci. Publ., Hackensack, NJ (2007), pp. 1-15 | MR 2334322 | Zbl 1149.46022

[2] Bayart, F. Linearity of sets of strange functions, Michigan Math. J., Tome 53 (2005) no. 2, pp. 291-303 | Article | MR 2152701 | Zbl 1092.46006

[3] Bayart, F.; Grosse-Erdmann, K.-G.; Nestoridis, V.; Papadimitropoulos, C. Abstract theory of universal series and applications, Proc. London Math. Soc., Tome 96 (2008), pp. 417-463 | Article | MR 2396846 | Zbl 1147.30003

[4] Bonet, J. A problem on the structure of Fréchet spaces, Rev. R. Acad. Cien. Serie A. Mat., Tome 104 (2010), pp. 427-434 | Article | MR 2757250 | Zbl 1262.46001

[5] Bonet, J.; Martìnez-Giménez, F.; Peris, A. Universal and chaotic multipliers on spaces of operators, J. Math. Anal. Appl., Tome 294 (2004), pp. 599-611 | Article | MR 2088683 | Zbl 1062.47011

[6] Borel, E. Sur quelques points de la théorie des fonctions, Ann. Sci. École Norm. Sup., Tome 12 (1895), pp. 9-55 | MR 1508908

[7] Bés, J.; Conejero, J. A. Hypercyclic subspaces in omega, J. Math. Anal. Appl., Tome 316 (2006), pp. 16-23 | Article | MR 2201746 | Zbl 1094.47011

[8] Charpentier, S. On the closed subspaces of universal series in Banach spaces and Fréchet spaces, Studia Math., Tome 198 (2010), pp. 121-145 | Article | MR 2640073 | Zbl 1201.30005

[9] Grosse Erdmann, K-G. Universal families and hypercyclic operators, Bull. Amer. Math. Soc. (N.S.), Tome 36 (1999), pp. 345-381 | Article | MR 1685272 | Zbl 0933.47003

[10] Komatsu, H. Ultradistributions, I. Structure theorems and a characterization, J. Fac. Sci. Univ. Tokyo, Sect. 1A, Tome 20 (1973), pp. 25-105 | MR 320743 | Zbl 0258.46039

[11] León Saavedra, F.; Müller, V. Hypercyclic sequences of operators, Studia Math., Tome 175 (2006), pp. 1-18 | Article | MR 2261697 | Zbl 1106.47011

[12] Menet, Q. Sous-espace fermés de séries universelles sur un espace de Fréchet, Studia Math., Tome 207 (2011), pp. 181-195 | Article | MR 2864388 | Zbl 1256.30053

[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., Tome 138 (2010) no. 11, pp. 3945-3955 | Article | MR 2679616 | Zbl 1207.30082

[15] Petersson, H. Hypercyclic subspaces for Fréchet space operators, J. Math. Anal. Appl., Tome 319 (2006), pp. 764-782 | Article | MR 2227937 | Zbl 1101.47006

[16] Petzsche, H-J. On E. Borel’s theorem, Math. Ann., Tome 282 (1988), pp. 299-313 | Article | Zbl 0633.46033

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

[18] Seleznev, A. I. On universal power series, Math. Sbornik N.S., Tome 28 (1951), pp. 453-460 | MR 41928 | Zbl 0043.29501

[19] Tsirivas, N. Simultaneous approximation by universal series, Math. Nachr., Tome 283 (2010), pp. 909-920 | Article | MR 2668431 | Zbl 1194.41028