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Hypercyclic and topologically mixing cosine functions on Banach spaces


Authors: Antonio Bonilla and Pedro J. Miana
Journal: Proc. Amer. Math. Soc. 136 (2008), 519-528
MSC (2000): Primary 47D09, 47A16
DOI: https://doi.org/10.1090/S0002-9939-07-09036-3
Published electronically: October 24, 2007
MathSciNet review: 2358492
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Abstract: Our first aim in this paper is to give sufficient conditions for the hypercyclicity and topological mixing of a strongly continuous cosine function. We apply these results to study the cosine function associated to translation groups. We also prove that every separable infinite dimensional complex Banach space admits a topologically mixing uniformly continuous cosine family.


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  • 1. S. I. Ansari, Existence of hypercyclic operators on topological vector spaces, J. Funct. Anal. 148 (1997), no. 2, 384-390. MR 1469346 (98h:47028a)
  • 2. W. Arendt, C. Batty, M. Hieber, and F. Neubrander, Vector-valued Laplace Transforms and Cauchy problems, Birkhäuser, 2001. MR 1886588 (2003g:47072)
  • 3. T. Bermúdez, A. Bonilla, J. A. Conejero, and A. Peris, Hypercyclic, topologically mixing and chaotic semigroups in Banach spaces, Studia Math. 170 (2005), no. 1, 57-75. MR 2142183 (2005m:47013)
  • 4. T. Bermúdez, A. Bonilla, and A. Martinón, On the existence of chaotic and hypercyclic semigroups in Banach spaces, Proc. Amer. Math. Soc. 131 (2003), no. 8, 2435-2441. MR 1974641 (2004b:47006)
  • 5. L. Bernal-González, On hypercyclic operators on Banach spaces, Proc. Amer. Math. Soc. 127 (1999), no. 4, 1003-1010. MR 1476119 (99f:47010)
  • 6. J. Bès and A. Peris, Hereditarily hypercyclic operators, J. Funct. Anal. 167 (1999), no. 1, 94-112. MR 1710637 (2000f:47012)
  • 7. J. Bonet and A. Peris, Hypercyclic operators on non-normable Fréchet spaces, J. Funct. Anal. 159 (1998), no. 2, 587-595. MR 1658096 (99k:47044)
  • 8. W. Desch, W. Schappacher, and G. F. Webb, Hypercyclic and chaotic semigroups of linear operators, Ergodic Theory and Dynamical Syst. 17 (1997), 1-27. MR 1468101 (98j:47083)
  • 9. K. G. Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. 36(1999), no. 3, 345-381. MR 1685272 (2000c:47001)
  • 10. F. Martínez-Giménez and A. Peris, Chaos for backward shift operators, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 12 (2002), no. 8, 1703-1715. MR 1927407 (2003h:47056)
  • 11. R. I. Ovsepian and A. Pe\lczynski, On the existence of a fundamental total and bounded biorthogonal sequence in every separable Banach space, and related constructions of uniformly bounded orthonormal systems in $ l^2$, Studia Math. 54 (1975), 149-159. MR 0394137 (52:14942)
  • 12. S. Rolewicz, On orbits of elements, Studia Math. 32 (1969), 17-22. MR 0241956 (39:3292)

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Additional Information

Antonio Bonilla
Affiliation: Departamento de Análisis Matemático, Universidad de La Laguna, 38271 La Laguna (Tenerife), Spain
Email: abonilla@ull.es

Pedro J. Miana
Affiliation: Departamento de Matemáticas, Universidad de Zaragoza, 50009 Zaragoza, Spain
Email: pjmiana@unizar.es

DOI: https://doi.org/10.1090/S0002-9939-07-09036-3
Keywords: Hypercyclic operators, topologically mixing operators, cosine functions, translation groups.
Received by editor(s): July 17, 2006
Published electronically: October 24, 2007
Additional Notes: The first author is supported by MEC and FEDER MTM2005-07347 and MEC (Accion special) MTM2006-26627-E
The second author is supported by Project MTM2004-03036, DGI-FEDER, of the MCYT, Spain, and Project E-64, D. G. Aragón, Spain.
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2007 American Mathematical Society

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