Typical operators admit common cyclic vectors
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- by Pavel Zorin-Kranich PDF
- Proc. Amer. Math. Soc. 141 (2013), 2371-2378 Request permission
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. $R>0$).References
- Shamim I. Ansari, Hypercyclic and cyclic vectors, J. Funct. Anal. 128 (1995), no. 2, 374–383. MR 1319961, DOI 10.1006/jfan.1995.1036
- F. Bayart and É. Matheron, How to get common universal vectors, Indiana Univ. Math. J. 56 (2007), no. 2, 553–580. MR 2317538, DOI 10.1512/iumj.2007.56.2863
- Frédéric Bayart and Étienne Matheron, Dynamics of linear operators, Cambridge Tracts in Mathematics, vol. 179, Cambridge University Press, Cambridge, 2009. MR 2533318, DOI 10.1017/CBO9780511581113
- George D. Birkhoff, Surface transformations and their dynamical applications, Acta Math. 43 (1922), no. 1, 1–119. MR 1555175, DOI 10.1007/BF02401754
- Tanja Eisner, A “typical” contraction is unitary, Enseign. Math. (2) 56 (2010), no. 3-4, 403–410. MR 2769030, DOI 10.4171/LEM/56-3-6
- Tanja Eisner and Tamás Mátrai, On typical properties of Hilbert space operators, to appear in Israel J. Math.
- 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
- Karl-Goswin Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 3, 345–381. MR 1685272, DOI 10.1090/S0273-0979-99-00788-0
- Paul R. Halmos, In general a measure preserving transformation is mixing, Ann. of Math. (2) 45 (1944), 786–792. MR 11173, DOI 10.2307/1969304
- Paul R. Halmos, A Hilbert space problem book, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0208368
- M. G. Nadkarni, Spectral theory of dynamical systems, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 1998. MR 1719722, DOI 10.1007/978-3-0348-8841-7
- V. Rohlin, A “general” measure-preserving transformation is not mixing, Doklady Akad. Nauk SSSR (N.S.) 60 (1948), 349–351 (Russian). MR 0024503
- Rebecca Sanders, Common hypercyclic vectors and the hypercyclicity criterion, Integral Equations Operator Theory 65 (2009), no. 1, 131–149. MR 2545665, DOI 10.1007/s00020-009-1711-0
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
- 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
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 2371-2378
- MSC (2010): Primary 47A16
- DOI: https://doi.org/10.1090/S0002-9939-2013-11512-1
- MathSciNet review: 3043018