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Rank One Perturbation of Unitary Operators with Full Measure of Hypercyclic Vectors

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Abstract

In this note, we find a class of unitary operators, denoted by \(\mathcal {U}\), on a complex separable infinite-dimensional Hilbert space \(\mathcal {H}\) such that for any \(U\in \mathcal {U}\), there exists an operator R of rank 1 on \(\mathcal {H}\) such that \(U+R\) is hypercyclic and the hypercyclic vectors are of full measure. Then, these results are applied to the controllability of discrete-time linear control systems, where the rank one perturbation is used as a one-dimensional feedback control law.

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References

  1. Ansari, S.: Existence of hypercyclic operators on topological vector spaces. J. Func. Anal. 148, 384–390 (1997)

    Article  MathSciNet  Google Scholar 

  2. Baranov, A., Lishanskii, A.: On S. Grivaux’ example of a hypercyclic rank one perturbation of a unitary operator. Arch. Math. (Basel) 104(3), 223–235 (2015)

  3. Baranov, A., Kapustin, V., Lishanskii, A.: On hypercyclic rank one perturbations of unitary operators. Math. Nachr. 292, 961–968 (2019)

    Article  MathSciNet  Google Scholar 

  4. Bayart, F., Grivaux, S.: Hypercyclicity and unimodular point spectrum. J. Funct. Anal. 226, 281–300 (2005)

    Article  MathSciNet  Google Scholar 

  5. Bayart, F., Grivaux, S.: Frequently hypercyclic operators. Trans. Am. Math. Soc. 358, 5083–5117 (2006)

    Article  MathSciNet  Google Scholar 

  6. Bayart, F., Matheron, É.: Dynamics of Linear Operators. Cambridge Tracts in Mathematics, vol. 179. Cambridge University Press, Cambridge (2009)

    Book  Google Scholar 

  7. Bernal-Gonzalez, L.: On hypercyclic operators on Banach spaces. Proc. Am. Math. Soc. 127, 1003–1010 (1999)

    Article  MathSciNet  Google Scholar 

  8. Curtain, R.F., Pritchard, A.J.: Infinite-Dimensional Linear Systems Theory. Lecture Notes in Control and Information Sciences, vol. 8. Springer, Berlin (1978)

    Book  Google Scholar 

  9. Curtain, R.F., Zwart, H.: An Introduction to Infinite-Dimensional Linear Systems Theory. Springer, New York (1995)

    Book  Google Scholar 

  10. Fattorini, H.O.: On control in finite time of differential equations in Banach space. Commun. Pure App. Math. XIX, 17–34 (1966)

    MathSciNet  MATH  Google Scholar 

  11. Fattorini, H.O.: On complete controllability of linear systems. J. Differ. Eqn. 3, 391–402 (1967)

    Article  MathSciNet  Google Scholar 

  12. Gethner, R.M., Shapiro, J.H.: Universal vectors for operators on spaces of holomorphic functions. Proc. Am. Math. Soc. 100, 281–288 (1987)

    Article  MathSciNet  Google Scholar 

  13. Grivaux, S.: A new class of frequently hypercyclic operators. Indiana Univ. Math. J. 60, 1177–1202 (2011)

    Article  MathSciNet  Google Scholar 

  14. Grivaux, S.: A hypercyclic rank one perturbation of a unitary operator. Math. Nachr. 285, 533–544 (2012)

    Article  MathSciNet  Google Scholar 

  15. Grosse-Erdmann, K.-G., Peris, A.: Linear Chaos. Springer, London (2011)

    Book  Google Scholar 

  16. Liu, X., Huang, Y., Xiao, M.: Approximately nearly controllability of discrete-time bilinear control systems with control input characteristic. J. Frankl. Inst. 352, 1561–1579 (2015)

    Article  MathSciNet  Google Scholar 

  17. Kitai, C.: Invariant closed sets for linear operators. Thesis, University of Toronto (1982)

  18. Rolewicz, S.: On orbits of elements. Studia Math. 32, 17–22 (1969)

    Article  MathSciNet  Google Scholar 

  19. Salas, H.: Supercyclicity and weighted shifts. Studia Math. 135, 55–74 (1999)

    Article  MathSciNet  Google Scholar 

  20. Schechter, S.: On the inversion of certain matrices. Math. Comput. 13, 73–77 (1959)

    Article  MathSciNet  Google Scholar 

  21. Shkarin, S.: A hypercyclic finite rank perturbation of a unitary operator. Math. Ann. 348, 379–393 (2010)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors wish to thank the anonymous referees for their careful reading and for valuable comments which improved the quality of the manuscript. The first author was supported by the National Natural Science Foundation of China (No. 12001113) and the Educational Commission of Guangdong Province (No. 2019KQNCX096). The second author was supported by National Natural Science Foundation of China (No. 11701584 and 11801096), the Natural Science Research Project of Guangdong Province (No. 2018KTSCX122 and 2017KQNCX122). The authors would like to thank Prof. Yu Huang and Prof. Haiwei Sun for very useful suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Guangdong University of Education, Guangzhou, 510303, People’s Republic of China

    Xin Liu

  2. School of Mathematics and Systems Science, Guangdong Polytechnic Normal University, Guangzhou, 510665, People’s Republic of China

    Zhijing Chen

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  1. Xin Liu
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  2. Zhijing Chen
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Correspondence to Xin Liu.

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Communicated by Pedro Tradacete.

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Liu, X., Chen, Z. Rank One Perturbation of Unitary Operators with Full Measure of Hypercyclic Vectors. Bull. Malays. Math. Sci. Soc. 45, 2475–2492 (2022). https://doi.org/10.1007/s40840-022-01334-9

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  • DOI: https://doi.org/10.1007/s40840-022-01334-9

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