On the set of hypercyclic vectors for the differentiation operator. (English) Zbl 1218.47017
Let \(\mathcal H(\mathbb C)\) be the space of entire functions endowed with the compact-open topology, and let \(D\) be the differentiation operator. G.R.MacLane [J. Anal.Math.2, 72–87 (1952; Zbl 0049.05603)] showed that \(D\) is hypercyclic. Let \(H(D)\) be the set of hypercyclic functions for \(D.\)
The author answers affirmatively two questions raised by R.M.Aron, J.A.Conejero, A.Peris and J.B.Seoane-Sepúlveda [Contemp.Math.435, 47–52 (2007; Zbl 1149.46024)]. He constructs a hypercyclic subspace for \(D\); i.e., an infinite-dimensional closed subspace \(M\) such that \(M\setminus \{0\}\subset H(D)\). He also constructs \(f \in \mathcal H(\mathbb C)\) whose generated algebra \(A\) satisfies that \(A\setminus \{0\}\subset H(D)\). This last construction is very intricate; but, as the author points out, the existence of such a function was also proved (independently) by F.Bayart and E.Matheron [“Dynamics of linear operators” (Cambridge Tracts in Mathematics 179) (2009; Zbl 1187.47001)].
The author answers affirmatively two questions raised by R.M.Aron, J.A.Conejero, A.Peris and J.B.Seoane-Sepúlveda [Contemp.Math.435, 47–52 (2007; Zbl 1149.46024)]. He constructs a hypercyclic subspace for \(D\); i.e., an infinite-dimensional closed subspace \(M\) such that \(M\setminus \{0\}\subset H(D)\). He also constructs \(f \in \mathcal H(\mathbb C)\) whose generated algebra \(A\) satisfies that \(A\setminus \{0\}\subset H(D)\). This last construction is very intricate; but, as the author points out, the existence of such a function was also proved (independently) by F.Bayart and E.Matheron [“Dynamics of linear operators” (Cambridge Tracts in Mathematics 179) (2009; Zbl 1187.47001)].
Reviewer: Héctor N. Salas (Mayagüez)
MSC:
| 47A16 | Cyclic vectors, hypercyclic and chaotic operators |
| 32A15 | Entire functions of several complex variables |
References:
| [1] | S. Ansari, Hypercyclic and cyclic vectors, Journal of Functionl Analysis 128 (1995), 374–383. · Zbl 0853.47013 · doi:10.1006/jfan.1995.1036 |
| [2] | R. Aron, J. Conejero, A. Peris and J. Seoane-Sepúlveda, Sums and Products of Bad Functions. Function Spaces, Contemporary Mathematics 435, American Mathematical Society, Providence, RI, 2007, pp. 47–52. |
| [3] | R. Aron, J. Conejero, A. Peris and J. Seoane-Sepúlveda, Powers of hypercyclic functions for some classical hypercyclic operators, Integral Equations Operator Theory 58 (2007), 591–596. · Zbl 1142.47009 · doi:10.1007/s00020-007-1490-4 |
| [4] | F. Bayart and E. Matheron, Dynamics of Linear Operators, Cambridge University Press, 2009. · Zbl 1187.47001 |
| [5] | P. Bourdon, Invariant manifolds of hypercyclic vectors, Proceedings of the American Mathematical Society 118 (1993), 845–847. · Zbl 0809.47005 · doi:10.1090/S0002-9939-1993-1148021-4 |
| [6] | M. Gonzáles, F. Leon-Saavedra and A. Montes-Rodríguez, Semi-Fredholm theory: hypercyclic and supercyclic subspaces, Proceedings of the London Mathematical Society 81 (2000), 169–189. · Zbl 1028.47007 · doi:10.1112/S0024611500012454 |
| [7] | K. Grosse-Erdmann, Universal families and hypercyclic operators, Bulletin of the American Mathematical Society 36 (1999), 345–381. · Zbl 0933.47003 · doi:10.1090/S0273-0979-99-00788-0 |
| [8] | K. Grosse-Erdmann, Recent developments in hypercyclicity, RACSAM. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 97 (2003), 273–286. |
| [9] | F. Leon-Saavedra and A. Montes-Rodríguez, Spectral theory and hypercyclic subspaces, Transactions of the American Mathematical Society 353 (1997), 247–267. · Zbl 0961.47003 · doi:10.1090/S0002-9947-00-02743-4 |
| [10] | G. MacLane, Sequences of derivatives and normal families, Journal d’Analyse Mathematique 2 (1952), 72–87. · Zbl 0049.05603 · doi:10.1007/BF02786968 |
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