×

Hypercyclic behavior of translation operators on spaces of analytic functions on Hilbert spaces. (English) Zbl 1343.47007

Hypercyclity is one of the main concepts in linear dynamics. An operator \(T:X \to X\) on a Fréchet space is hypercyclic if there exists a vector \(x \in X\) whose orbit \(\{x, Tx, T^{2}x, \ldots \}\) is dense in \(X\). A result by G. D. Birkhoff in [C. R. Acad. Sci., Paris 189, 473–475 (1929; JFM 55.0192.07)] shows that the translation operator \(T_{a} : f(z) \mapsto f(z+a)\) (with \(a \in \mathbb{C}\)) from the space of entire functions \(H(\mathbb{C})\) into itself is hypercyclic. Since then, the hypercyclity of this operator on different spaces has received the attention of several mathematicians.
The present paper goes on this direction, considering the translation operator on a certain Hilbert space: a generalized symmetric Fock space. This is defined from a Hilbert space \(E\) as the Hilbert direct sum (with some Hilbert norm \(\| \cdot \|_{\eta}\)) of the symmetric tensor powers \[ \mathcal{F}_{\eta} = \mathbb{C} \oplus E \oplus \otimes_{s}^{2}E \oplus \otimes_{s}^{3}E \oplus \cdots \] Conditions are given so that the space \(\mathcal{H}_{\eta}= \mathcal{F}_{\eta}^{*}\) consists of entire functions on \(E\).
Then the main result of the paper analyzes when the translation operator \(T_{a} : \mathcal{H}_{\eta} \to \mathcal{H}_{\eta}\) (with \(a \in E\)) is hypercyclic. As a preceding result, a characterization for the differentiation operator \(D_{a}\) to be bounded on \(\mathcal{H}_{\eta}\) is given.

MSC:

47A16 Cyclic vectors, hypercyclic and chaotic operators
47B33 Linear composition operators
46E50 Spaces of differentiable or holomorphic functions on infinite-dimensional spaces

Citations:

JFM 55.0192.07
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Grosse-Erdmann, K.-G., Universal families and hypercyclic operators, Bulletin of the American Mathematical Society. New Series, 36, 3, 345-381 (1999) · Zbl 0933.47003 · doi:10.1090/s0273-0979-99-00788-0
[2] Birkhoff, G. D., D’émonstration d’un théorème élémentaire sur es fonctions entières, Comptes Rendus de l’Académie des Sciences de Paris, 189, 473-475 (1929) · JFM 55.0192.07
[3] MacLane, G. R., Sequences of derivatives and normal families, Journal d’Analyse Mathématique, 2, 72-87 (1952) · Zbl 0049.05603 · doi:10.1007/bf02786968
[4] Bayart, F.; Matheron, E., Dynamics of Linear Operators. Dynamics of Linear Operators, Cambridge Tracts in Mathematics, 179 (2009), New York, NY, USA: Cambridge University Press, New York, NY, USA · Zbl 1187.47001 · doi:10.1017/cbo9780511581113
[5] Grosse-Erdmann, K.-G.; Peris Manguillot, A., Linear Chaos. Linear Chaos, Universitext (2011), London, UK: Springer, London, UK · Zbl 1246.47004 · doi:10.1007/978-1-4471-2170-1
[6] Godefroy, G.; Shapiro, J. H., Operators with dense, invariant, cyclic vector manifolds, Journal of Functional Analysis, 98, 2, 229-269 (1991) · Zbl 0732.47016 · doi:10.1016/0022-1236(91)90078-J
[7] Aron, R.; Bès, J., Hypercyclic differentiation operators, Contemporary Mathematics, 232, 39-46 (1999) · Zbl 0938.47004
[8] Novosad, Z.; Zagorodnyuk, A., Polynomial automorphisms and hypercyclic operators on spaces of analytic functions, Archiv der Mathematik, 89, 2, 157-166 (2007) · Zbl 1137.47009 · doi:10.1007/s00013-007-2043-4
[9] Chan, K. C.; Shapiro, J. H., The cyclic behavior of translation operators on Hilbert spaces of entire functions, Indiana University Mathematics Journal, 40, 4, 1421-1449 (1991) · Zbl 0771.47015 · doi:10.1512/iumj.1991.40.40064
[10] Kitai, C., Invariant closed sets for linear operators [Ph.D. thesis] (1982), University of Toronto
[11] Gethner, R. M.; Shapiro, J. H., Universal vectors for operators on spaces of holomorphic functions, Proceedings of the American Mathematical Society, 100, 2, 281-288 (1987) · Zbl 0618.30031 · doi:10.2307/2045959
[12] Dineen, S., Complex Analysis on Infinite Dimensional Spaces. Complex Analysis on Infinite Dimensional Spaces, Monographs in Mathematics (1999), New York, NY, USA: Springer, New York, NY, USA · Zbl 1034.46504 · doi:10.1007/978-1-4471-0869-6
[13] Mujica, J., Complex Analysis in Banach Spaces. Complex Analysis in Banach Spaces, North-Holland Mathematics Studies, 120 (1986), Amsterdam, The Netherlands: North-Holland, Amsterdam, The Netherlands · Zbl 0586.46040
[14] Lopushansky, O. V.; Zagorodnyuk, A. V., A Hilbert space of functions of infinitely many variables, Methods of Functional Analysis and Topology, 10, 2, 13-20 (2004) · Zbl 1056.46033
[15] Lopushansky, O.; Zagorodnyuk, A., Infinite Dimensional Holomorphy. Spectra and Hilbertian Structures (2013), Kraków, Poland: AGH University of Science and Technology Press, Kraków, Poland
[16] Saitoh, S., Integral Transforms, Reproducing Kernels and Their Applications. Integral Transforms, Reproducing Kernels and Their Applications, Pitman Research Notes in Mathematics Series, 369 (1997), Harlow, UK: Longman, Harlow, UK · Zbl 0891.44001
[17] Arcozzi, N.; Rochberg, R.; Sawyer, E., The diameter spaces, a restriction of the Drury-Arveson-Hardy space, Contemporary Mathematics, 435, 21-42 (2007) · Zbl 1149.46023
[18] Cole, B. J.; Gamelin, T. W., Representing measures and Hardy spaces for the infinite polydisk algebra, Proceedings of the London Mathematical Society, 53, 1, 112-142 (1986) · Zbl 0624.46032 · doi:10.1112/plms/s3-53.1.112
[19] Lopushansky, O., Best approximations in Hardy spaces on infinite-dimensional unitary matrix groups, Abstract and Applied Analysis, 2014 (2014) · Zbl 1474.43012 · doi:10.1155/2014/631503
[20] Lopushansky, O.; Zagorodnyuk, A., Representing measures and infinite-dimensional holomorphy, Journal of Mathematical Analysis and Applications, 333, 2, 614-625 (2007) · Zbl 1131.46033 · doi:10.1016/j.jmaa.2006.09.035
[21] Lopushansky, O.; Zagorodnyuk, A., Hardy type spaces associated with compact unitary groups, Nonlinear Analysis: Theory, Methods & Applications, 74, 2, 556-572 (2011) · Zbl 1210.46029 · doi:10.1016/j.na.2010.09.009
[22] Neeb, K.-H.; Ørsted, B.; Doebrer, H.-D.; Dobrev, V. K.; Hilgert, J., Hardy spaces in an infinite dimensional setting, Proceedings of the 2nd International Workshop on “Lie Thery and Its Application in Physics”
[23] Mozhyrovska, Z. H.; Zagorodnyuk, A. V., Hypercyclic composition operators on Hilbert spaces of analytic functions, Methods of Functional Analysis and Topology, 20, 3, 284-291 (2014) · Zbl 1324.47017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.