Hypercyclicity of convolution operators on spaces of entire functions
[Hypercyclicité d’opérateurs de convolution sur des espaces de fonctions entières]
Annales de l'Institut Fourier, Tome 63 (2013) no. 4, pp. 1263-1283.

Dans cet article, nous utilisons les types d’holomorphie de Nachbin pour généraliser certains résultats récents concernant les opérateurs de convolutions hypercycliques sur les espaces de Fréchet de fonctions d’un nombre infini de variables complexes, entières, de type borné.

In this paper we use Nachbin’s holomorphy types to generalize some recent results concerning hypercyclic convolution operators on Fréchet spaces of entire functions of bounded type of infinitely many complex variables

DOI : 10.5802/aif.2803
Classification : 32DXX, 47A16, 46G20
Keywords: Fréchet spaces of entire functions, hypercyclicity, convolution operators
Mot clés : Espaces de Fréchet de fonctions entières, hypercyclicité, opérateurs de convolution
Bertoloto, F.J. 1 ; Botelho, G. 1 ; Fávaro, V.V. 1 ; Jatobá, A.M. 1

1 Universidade Federal de Uberlândia Faculdade de Matemática 38.400-902 - Uberlândia (Brazil)
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     title = {Hypercyclicity of convolution operators on spaces of entire functions},
     journal = {Annales de l'Institut Fourier},
     pages = {1263--1283},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Bertoloto, F.J.; Botelho, G.; Fávaro, V.V.; Jatobá, A.M. Hypercyclicity of convolution operators on spaces of entire functions. Annales de l'Institut Fourier, Tome 63 (2013) no. 4, pp. 1263-1283. doi : 10.5802/aif.2803. http://archive.numdam.org/articles/10.5802/aif.2803/

[1] Aron, Richard; Bès, Juan Hypercyclic differentiation operators, Function spaces (Edwardsville, IL, 1998) (Contemp. Math.), Volume 232, Amer. Math. Soc., Providence, RI, 1999, pp. 39-46 | DOI | MR | Zbl

[2] Aron, Richard; Markose, Dinesh On universal functions, J. Korean Math. Soc., Volume 41 (2004) no. 1, pp. 65-76 (Satellite Conference on Infinite Dimensional Function Theory) | DOI | MR | Zbl

[3] Birkhoff, G. D. Démonstration d’un théorème élémentaire sur les fonctions entières, C. R. Acad. Sci. Paris, Volume 189 (1929), pp. 473-475

[4] Botelho, Geraldo; Braunss, H.-A.; Junek, H.; Pellegrino, Daniel M. Holomorphy types and ideals of multilinear mappings, Studia Math., Volume 177 (2006) no. 1, pp. 43-65 | DOI | MR | Zbl

[5] Botelho, Geraldo; Pellegrino, Daniel M. Two new properties of ideals of polynomials and applications, Indag. Math. (N.S.), Volume 16 (2005) no. 2, pp. 157-169 | DOI | MR | Zbl

[6] Carando, Daniel; Dimant, Verónica; Muro, Santiago Hypercyclic convolution operators on Fréchet spaces of analytic functions, J. Math. Anal. Appl., Volume 336 (2007) no. 2, pp. 1324-1340 | DOI | MR | Zbl

[7] Carando, Daniel; Dimant, Verónica; Muro, Santiago Coherent sequences of polynomial ideals on Banach spaces, Math. Nachr., Volume 282 (2009) no. 8, pp. 1111-1133 | DOI | MR | Zbl

[8] Carando, Daniel; Dimant, Verónica; Muro, Santiago Every Banach ideal of polynomials is compatible with an operator ideal, Monatsh. Math., Volume 165 (2012) no. 1, pp. 1-14 | DOI | MR | Zbl

[9] Dineen, Seán Complex analysis on infinite-dimensional spaces, Springer Monographs in Mathematics, Springer-Verlag London Ltd., London, 1999 | DOI | MR | Zbl

[10] Fávaro, Vinícius V. Convolution equations on spaces of quasi-nuclear functions of a given type and order, Bull. Belg. Math. Soc. Simon Stevin, Volume 17 (2010) no. 3, pp. 535-569 http://projecteuclid.org/getRecord?id=euclid.bbms/1284570737 | MR | Zbl

[11] Fávaro, Vinícius V.; Jatobá, Ariosvaldo M. Holomorphy types and spaces of entire functions of bounded type on Banach spaces, Czechoslovak Math. J., Volume 59(134) (2009) no. 4, pp. 909-927 | DOI | MR | Zbl

[12] Gámez-Merino, José L.; Muñoz-Fernández, Gustavo A.; Sánchez, Víctor M.; Seoane-Sepúlveda, Juan B. Sierpiński-Zygmund functions and other problems on lineability, Proc. Amer. Math. Soc., Volume 138 (2010) no. 11, pp. 3863-3876 | DOI | MR | Zbl

[13] Gethner, Robert M.; Shapiro, Joel H. Universal vectors for operators on spaces of holomorphic functions, Proc. Amer. Math. Soc., Volume 100 (1987) no. 2, pp. 281-288 | DOI | MR | Zbl

[14] Godefroy, Gilles; Shapiro, Joel H. Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal., Volume 98 (1991) no. 2, pp. 229-269 | DOI | MR | Zbl

[15] Gupta, Chaitan P. Convolution operators and holomorphic mappings on a Banach space, Séminaire d’Analyse Moderne, No. 2, Dept. Math, Université de Sherbrooke, Québec, 1969 | Zbl

[16] Gupta, Chaitan P. On the Malgrange theorem for nuclearly entire functions of bounded type on a Banach space, Nederl. Akad. Wetensch. Proc. Ser. A73 = Indag. Math., Volume 32 (1970), pp. 356-358 | DOI | MR | Zbl

[17] Hallack, André Arbex Hypercyclicity for translations through Runge’s theorem, Bull. Korean Math. Soc., Volume 44 (2007) no. 1, pp. 117-123 | DOI | MR | Zbl

[18] Kitai, Carol Invariant closed sets for linear operators, ProQuest LLC, Ann Arbor, MI, 1982 Thesis (Ph.D.)–University of Toronto (Canada) | MR

[19] MacLane, G. R. Sequences of derivatives and normal families, J. Analyse Math., Volume 2 (1952), pp. 72-87 | DOI | MR | Zbl

[20] Malgrange, Bernard Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution, Ann. Inst. Fourier, Grenoble, Volume 6 (1955–1956), pp. 271-355 | DOI | Numdam | MR | Zbl

[21] Matos, Mário C. Mappings between Banach spaces that send mixed summable sequences into absolutely summable sequences, J. Math. Anal. Appl., Volume 297 (2004) no. 2, pp. 833-851 (Special issue dedicated to John Horváth) | DOI | MR | Zbl

[22] Matos, Mário C. Absolutely Summing Mappings, Nuclear Mappings and Convolution Equations, IMECC-UNICAMP, 2005 (http://www.ime.unicamp.br/rel_pesq/2007/pdf/rp03-07.pdf)

[23] Mujica, Jorge Complex analysis in Banach spaces, North-Holland Mathematics Studies, 120, North-Holland Publishing Co., Amsterdam, 1986 | MR | Zbl

[24] Mujica, X. Aplicações τ ( p ; q ) -somantes e σ ( p ) -nucleares, Universidade Estadual de Campinas (2006) (Ph. D. Thesis http://cutter.unicamp.br/document/?code =vtls000378266)

[25] Nachbin, Leopoldo Topology on spaces of holomorphic mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 47, Springer-Verlag New York Inc., New York, 1969 | MR | Zbl

[26] Petersson, Henrik Hypercyclic convolution operators on entire functions of Hilbert-Schmidt holomorphy type, Ann. Math. Blaise Pascal, Volume 8 (2001) no. 2, pp. 107-114 | DOI | Numdam | MR | Zbl

[27] Petersson, Henrik Hypercyclic subspaces for Fréchet space operators, J. Math. Anal. Appl., Volume 319 (2006) no. 2, pp. 764-782 | DOI | MR | Zbl

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