A hypercyclic composition operator on a Hilbert space of entire functions

Z.H.Novosad; A.V.Zagorodnyuk

Let $E$ be a separable Hilbert space. We consider a special Hilbert space of entire functions on $E$, ${\cal H}^2(E)$, and show that the operator of composition with translation $x\mapsto x+a$ is hypercyclic in ${\cal H}^2(E).$

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7. Lopushansky O. and Zagorodnyuk A. Hilbert spaces of functions of infinitely many variables, Methods Funct. Anal. Topology 10, no 2 (2004), 13--20.

8. Neeb K.- H. Holomorphy and Convexity in Lie Theory, De Gruyter Expositions in Mathematics, Vol.~28, Berlin, New York, 2000.

9. Neeb K.- H. and rsted B. Hardy Spaces in an Infinite Dimensional Setting, Proceedings of II international workshop ``Lie theory and its application in physics'', Clausthal, Germany (ed. H.-D. Doebrer, V.K. Dobrev and J. Hilgert), 1988, 3--27.

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	A hypercyclic composition operator on a Hilbert space of entire functions
Author	Z.H.Novosad, A.V.Zagorodnyuk Dept. of Functional Analysis, Institute of Applied Problems of Mechanics and Mathematics, 3b Str. Naukova, Lviv 79060, Ukraine
Abstract	Let $E$ be a separable Hilbert space. We consider a special Hilbert space of entire functions on $E$, ${\cal H}^2(E)$, and show that the operator of composition with translation $x\mapsto x+a$ is hypercyclic in ${\cal H}^2(E).$
Keywords	hypercyclic operator, Hilbert space, entire function
DOI	doi:10.30970/ms.23.1.108-112
Reference	1. Aron R., Bès J. Hypercyclic differentiation operators, Contemporary Mathematics, 232 (1999), 39-46. 2. Birkhoff G. D. Démonstration d'un théorème élémentaire sur les fonctions entières, C. R. Acad. Sci. Paris 189 (1929), 473-475. 3. Bonet J. and Peris A. Hypercyclic operators on non-normable Fréchet space, J. Funct. Anal. 159 (1998), 387–395. 4. Chan K.C. and Shapiro J.H. The cyclic behavior of translation operators on Hilbert spaces of entire functions, Indiana Univ. Math. J. 40 (1991), 1421--1449. 5. Dineen S. Complex Analysis on Infinite Dimensional Spaces, Monographs in Mathematics, Springer, New York, 1999. 6. Grosse-Erdmann K.-G. Universal families and hypercyclic operators, Bull. Amer. Math. Soc. (N.S.) 36 (1999), 345--381. 7. Lopushansky O. and Zagorodnyuk A. Hilbert spaces of functions of infinitely many variables, Methods Funct. Anal. Topology 10, no 2 (2004), 13--20. 8. Neeb K.- H. Holomorphy and Convexity in Lie Theory, De Gruyter Expositions in Mathematics, Vol.~28, Berlin, New York, 2000. 9. Neeb K.- H. and rsted B. Hardy Spaces in an Infinite Dimensional Setting, Proceedings of II international workshop ``Lie theory and its application in physics'', Clausthal, Germany (ed. H.-D. Doebrer, V.K. Dobrev and J. Hilgert), 1988, 3--27. 10. Read C.J. The invariant subspace problem for a class of Banach spaces. II. Hypercyclic operators, Israel J. Math. 63 (1988), 1--40.
Pages	108-112
Volume	23
Issue	1
Year	2005
Journal	Matematychni Studii
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