Hypercyclic algebras for 𝐷-multiples of convolution operators

Bernal-González, Luis; Calderón-Moreno, María del Carmen

doi:10.1090/proc/14146

Hypercyclic algebras for $D$-multiples of convolution operators
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by Luis Bernal-González and María del Carmen Calderón-Moreno PDF

Proc. Amer. Math. Soc. 147 (2019), 647-653 Request permission

Abstract:

It is shown in this short note the existence, for each nonzero member of the ideal of $D$-multiples of convolution operators acting on the space of entire functions, of a scalar multiple of it supporting a hypercyclic algebra.

References

Lars V. Ahlfors, Complex analysis, 3rd ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York, 1978. An introduction to the theory of analytic functions of one complex variable. MR 510197
Richard M. Aron, Luis Bernal González, Daniel M. Pellegrino, and Juan B. Seoane Sepúlveda, Lineability: the search for linearity in mathematics, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2016. MR 3445906
R. M. Aron, J. A. Conejero, A. Peris, and J. B. Seoane-Sepúlveda, Powers of hypercyclic functions for some classical hypercyclic operators, Integral Equations Operator Theory 58 (2007), no. 4, 591–596. MR 2329137, DOI 10.1007/s00020-007-1490-4
R. M. Aron, J. A. Conejero, A. Peris, and J. B. Seoane-Sepúlveda, Sums and products of bad functions, Function spaces, Contemp. Math., vol. 435, Amer. Math. Soc., Providence, RI, 2007, pp. 47–52. MR 2359417, DOI 10.1090/conm/435/08365
Frédéric Bayart and Étienne Matheron, Dynamics of linear operators, Cambridge Tracts in Mathematics, vol. 179, Cambridge University Press, Cambridge, 2009. MR 2533318, DOI 10.1017/CBO9780511581113
Carlos A. Berenstein and Roger Gay, Complex analysis and special topics in harmonic analysis, Springer-Verlag, New York, 1995. MR 1344448, DOI 10.1007/978-1-4613-8445-8
L. Bernal-González, J.A. Conejero, G. Costakis, and J.A. Seoane-Sepúlveda, Multiplicative structures of hypercyclic functions for MacLane’s operator, J. Operator Theory (2017), in press, available at arXiv1710.10413v1 [mathFA].
Juan P. Bès, Invariant manifolds of hypercyclic vectors for the real scalar case, Proc. Amer. Math. Soc. 127 (1999), no. 6, 1801–1804. MR 1485460, DOI 10.1090/S0002-9939-99-04720-6
Juan Bès, J. Alberto Conejero, and Dimitris Papathanasiou, Convolution operators supporting hypercyclic algebras, J. Math. Anal. Appl. 445 (2017), no. 2, 1232–1238. MR 3545235, DOI 10.1016/j.jmaa.2016.01.029
J. Bès, J.A. Conejero, and D. Papathanasiou, Hypercyclic algebras for convolution and composition operators, Preprint (2017), available at arXiv:1706.08022v1 [math.FA].
J. Bès, J.A. Conejero, and D. Papathanasiou, Hypercyclic algebras for convolution and composition operators, Preprint (2018), available at arXiv:1706.08022v2 [math.FA].
J. Bès and D. Papathanasiou, Algebrable sets of hypercyclic vectors for convolution operators, Preprint (2017), available at arXiv:1706.08651v2 [math.FA].
G.D. Birkhoff, Démonstration d’un théorème élémentaire sur les fonctions entières, C. R. Acad. Sci. Paris 189 (1929), 473–475.
Paul S. Bourdon, Invariant manifolds of hypercyclic vectors, Proc. Amer. Math. Soc. 118 (1993), no. 3, 845–847. MR 1148021, DOI 10.1090/S0002-9939-1993-1148021-4
Leon Ehrenpreis, Mean periodic functions. I. Varieties whose annihilator ideals are principal, Amer. J. Math. 77 (1955), 293–328. MR 70047, DOI 10.2307/2372533
Gilles Godefroy and Joel H. Shapiro, Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal. 98 (1991), no. 2, 229–269. MR 1111569, DOI 10.1016/0022-1236(91)90078-J
Mario O. González, Complex analysis, Monographs and Textbooks in Pure and Applied Mathematics, vol. 152, Marcel Dekker, Inc., New York, 1992. Selected topics. MR 1153410
Karl-G. Grosse-Erdmann and Alfredo Peris Manguillot, Linear chaos, Universitext, Springer, London, 2011. MR 2919812, DOI 10.1007/978-1-4471-2170-1
Domingo A. Herrero, Limits of hypercyclic and supercyclic operators, J. Funct. Anal. 99 (1991), no. 1, 179–190. MR 1120920, DOI 10.1016/0022-1236(91)90058-D
A. S. B. Holland, Introduction to the theory of entire functions, Pure and Applied Mathematics, Vol. 56, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1973. MR 0447572
G. R. MacLane, Sequences of derivatives and normal families, J. Analyse Math. 2 (1952), 72–87 (English, with Hebrew summary). MR 53231, DOI 10.1007/BF02786968
Bernard Malgrange, Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution, Ann. Inst. Fourier (Grenoble) 6 (1955/56), 271–355 (French). MR 86990
Quentin Menet, Hypercyclic subspaces and weighted shifts, Adv. Math. 255 (2014), 305–337. MR 3167484, DOI 10.1016/j.aim.2014.01.012
Henrik Petersson, Hypercyclic subspaces for Fréchet space operators, J. Math. Anal. Appl. 319 (2006), no. 2, 764–782. MR 2227937, DOI 10.1016/j.jmaa.2005.06.042
Stanislav Shkarin, On the set of hypercyclic vectors for the differentiation operator, Israel J. Math. 180 (2010), 271–283. MR 2735066, DOI 10.1007/s11856-010-0104-z
Jochen Wengenroth, Hypercyclic operators on non-locally convex spaces, Proc. Amer. Math. Soc. 131 (2003), no. 6, 1759–1761. MR 1955262, DOI 10.1090/S0002-9939-03-07003-5