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Typical operators admit common cyclic vectors


Author: Pavel Zorin-Kranich
Journal: Proc. Amer. Math. Soc. 141 (2013), 2371-2378
MSC (2010): Primary 47A16
DOI: https://doi.org/10.1090/S0002-9939-2013-11512-1
Published electronically: March 14, 2013
MathSciNet review: 3043018
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Abstract: Given a countable dense subset $ D$ of an infinite-dimensional separable Hilbert space $ H$, the set of operators for which every vector in $ D$ except zero is hypercyclic (weakly supercyclic) is residual for the strong (resp. weak) operator topology in the unit ball of $ L(H)$ multiplied by $ R>1$ (resp. $ R>0$).


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  • [1] Shamim I. Ansari, Hypercyclic and cyclic vectors, J. Funct. Anal. 128 (1995), no. 2, 374-383. MR 1319961 (96h:47002)
  • [2] F. Bayart and É. Matheron, How to get common universal vectors, Indiana Univ. Math. J. 56 (2007), no. 2, 553-580. MR 2317538 (2008f:47008)
  • [3] Frédéric Bayart and Étienne Matheron, Dynamics of linear operators, Cambridge Tracts in Mathematics, vol. 179, Cambridge University Press, Cambridge, 2009. MR 2533318
  • [4] George D. Birkhoff, Surface transformations and their dynamical applications, Acta Math. 43 (1922), no. 1, 1-119. MR 1555175
  • [5] Tanja Eisner, A ``typical'' contraction is unitary, Enseign. Math. (2) 56 (2010), no. 3-4, 403-410. MR 2769030
  • [6] Tanja Eisner and Tamás Mátrai, On typical properties of Hilbert space operators, to appear in Israel J. Math.
  • [7] Tanja Eisner and András Serény, Category theorems for stable operators on Hilbert spaces, Acta Sci. Math. (Szeged) 74 (2008), no. 1-2, 259-270. MR 2431104 (2009i:47027)
  • [8] Karl-Goswin Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 3, 345-381. MR 1685272 (2000c:47001)
  • [9] Paul R. Halmos, In general a measure preserving transformation is mixing, Ann. of Math. (2) 45 (1944), 786-792. MR 0011173 (6,131d)
  • [10] Paul R. Halmos, A Hilbert space problem book, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0208368 (34:8178)
  • [11] M. G. Nadkarni, Spectral theory of dynamical systems, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 1998. MR 1719722 (2001d:37001)
  • [12] V. Rohlin, A ``general'' measure-preserving transformation is not mixing, Doklady Akad. Nauk SSSR (N.S.) 60 (1948), 349-351. MR 0024503 (9,504d)
  • [13] Rebecca Sanders, Common hypercyclic vectors and the hypercyclicity criterion, Integral Equations Operator Theory 65 (2009), no. 1, 131-149. MR 2545665 (2010i:47019)

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Additional Information

Pavel Zorin-Kranich
Affiliation: Korteweg de Vries Instituut voor Wiskunde, Universiteit van Amsterdam, P. O. Box 94248, 1090 GE Amsterdam, The Netherlands
Email: zorin-kranich@uva.nl

DOI: https://doi.org/10.1090/S0002-9939-2013-11512-1
Keywords: Cyclic vector, hypercyclic vector, weakly supercyclic vector, typical operator
Received by editor(s): August 26, 2010
Received by editor(s) in revised form: March 26, 2011, and October 18, 2011
Published electronically: March 14, 2013
Communicated by: Marius Junge
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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