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On the spectrum of frequently hypercyclic operators

Author: Stanislav Shkarin
Journal: Proc. Amer. Math. Soc. 137 (2009), 123-134
MSC (2000): Primary 47A16, 37A25
Published electronically: August 28, 2008
MathSciNet review: 2439433
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Abstract: A bounded linear operator $ T$ on a Banach space $ X$ is called frequently hypercyclic if there exists $ x\in X$ such that the lower density of the set $ \{n\in\mathbb{N}:T^nx\in U\}$ is positive for any non-empty open subset $ U$ of $ X$. Bayart and Grivaux have raised a question whether there is a frequently hypercyclic operator on any separable infinite dimensional Banach space. We prove that the spectrum of a frequently hypercyclic operator has no isolated points. It follows that there are no frequently hypercyclic operators on all complex and on some real hereditarily indecomposable Banach spaces, which provides a negative answer to the above question.

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  • 1. S. Ansari, Existence of hypercyclic operators on topological vector spaces, J. Funct. Anal. 148(1997), 384-390. MR 1469346 (98h:47028a)
  • 2. S. Ansari, Hypercyclic and cyclic vectors, J. Funct. Anal. 128(1995), 374-383. MR 1319961 (96h:47002)
  • 3. S. Argyros and A. Manoussakis, An indecomposable and unconditionally saturated Banach space, Studia Math. 159(2003), 1-32. MR 2030739 (2005d:46022)
  • 4. S. Argyros and A. Tolias, Methods in the theory of hereditarily indecomposable Banach spaces, Mem. Amer. Math. Soc. 170(2004), vi + 114 pp. MR 2053392 (2005f:46022)
  • 5. F. Bayart and S. Grivaux, Frequently hypercyclic operators, Trans. Amer. Math. Soc. 358(2006), 5083-5117. MR 2231886 (2007e:47013)
  • 6. L. Bernal-González, On hypercyclic operators on Banach spaces, Proc. Amer. Math. Soc. 127(1999), 1003-1010. MR 1476119 (99f:47010)
  • 7. J. Bés and A. Peris, Hereditarily hypercyclic operators, J. Funct. Anal. 167(1999), 94-112. MR 1710637 (2000f:47012)
  • 8. J. Bonet and A. Peris, Hypercyclic operators on non-normable Fréchet spaces, J. Funct. Anal. 159 (1998), 587-595. MR 1658096 (99k:47044)
  • 9. A. Bonilla and K.-G. Grosse-Erdmann, On a theorem of Godefroy and Shapiro, Integral Equations Operator Theory 56(2006), 151-162. MR 2264513 (2007h:47020)
  • 10. P. Bourdon and N. Feldman, Somewhere dense orbits are everywhere dense, Indiana Univ. Math. J. 52(2003), 811-819. MR 1986898 (2004d:47020)
  • 11. M. De La Rosa and C. Read, A hypercyclic operator whose direct sum is not hypercyclic, preprint.
  • 12. V. Ferenczi, Operators on subspaces of hereditarily indecomposable Banach spaces, Bull. London Math. Soc. 29(1997), 338-344. MR 1435570 (98b:47028)
  • 13. V. Ferenczi, Uniqueness of complex structure and real hereditarily indecomposable Banach spaces, Advances Math. 213(2007), 462-488. MR 2331251
  • 14. W. Gowers and B. Maurey, The unconditional basic sequence problem, J. Amer. Math. Soc. 6(1993), 851-874. MR 1201238 (94k:46021)
  • 15. S. Grivaux, Hypercyclic operators, mixing operators and the bounded steps problem, J. Operator Theory 54 (2005), 147-168. MR 2168865 (2006k:47021)
  • 16. S. Grivaux and S. Shkarin, Non-mixing hypercyclic operators, preprint.
  • 17. K.-G. Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. 36(1999), 345-381. MR 1685272 (2000c:47001)
  • 18. K.-G. Grosse-Erdmann, Recent developments in hypercyclicity, RACSAM Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. 97(2003), 273-286. MR 2068180 (2005c:47010)
  • 19. K.-G. Grosse-Erdmann and A. Peris, Frequently dense orbits, C. R. Math. Acad. Sci. Paris 341(2005), 123-128. MR 2153969 (2006a:47017)
  • 20. D. Herrero, Limits of hypercyclic and supercyclic operators, J. Funct. Anal. 99(1991), 179-190. MR 1120920 (92g:47026)
  • 21. D. Herrero and Z. Wang, Compact perturbations of hypercyclic and supercyclic operators, Indiana Univ. Math. J. 39(1990), 819-829. MR 1078739 (91k:47042)
  • 22. F. León-Saavedra and Vladimir Müller, Rotations of hypercyclic and supercyclic operators, Integral Equations and Operator Theory 50(2004), 385-391. MR 2104261 (2005g:47009)
  • 23. B. Levin, Distribution of zeros of entire functions, Amer. Math. Soc., Providence, Rhode Island, 1980. MR 589888 (81k:30011)
  • 24. A. Pietsch, Operator ideals, North-Holland, Amsterdam, 1980. MR 582655 (81j:47001)
  • 25. S. Shkarin, The Kitai criterion and backward shifts, Proc. Amer. Math. Soc. 136 (2008), 1659-1670. MR 2373595
  • 26. J. Wengenroth, Hypercyclic operators on non-locally convex spaces, Proc. Amer. Math. Soc. 131(2003), 1759-1761. MR 1955262 (2003j:47007)

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Additional Information

Stanislav Shkarin
Affiliation: Department of Pure Mathematics, Queens’s University Belfast, University Road, Belfast, BT7 1NN, United Kingdom

Keywords: Frequently hypercyclic operators, hereditarily indecomposable Banach spaces, quasinilpotent operators
Received by editor(s): July 26, 2007
Published electronically: August 28, 2008
Additional Notes: Partially supported by Plan Nacional I+D+I grant No. MTM2006-09060 and Junta de Andalucía FQM-260.
Communicated by: Marius Junge
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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