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A Few Remarks on Supercyclicity of Non-Archimedean Linear Operators on \(c_0(\mathbb N)\)

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Abstract

In this paper, we study the hypercyclic, supercyclic and cyclic properties of operators of the form \(I+B_{\bf b}\), where \(B_{\bf b}\) is a weighted backward shift defined on \(c_0(\mathbb N)\). The results are totally different from the real case.

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References

  1. V. Anashin and A. Khrennikov, Applied Algebraic Dynamics (Walter de Gruyter, Berlin, New York, 2009).

    Book  Google Scholar 

  2. A. El Amrani, A. Razouki, R. A. Hassani and M. Babahmed, “On the \(K\)-vector sequential topology on a non-Archimedean valued field,” p-Adic Num. Ultrametr. Anal. Appl. 12, 177–184 (2020).

    Article  MathSciNet  Google Scholar 

  3. M. Babahmed and A. El Asri, “Invariant subspace problem and compact operators on non-Archimedean Banach spaces,” Extr. Math. 35, 205–219 (2020).

    Google Scholar 

  4. F. Bayart and E. Matheron, Dynamics of Linear Operators (Cambridge Univ. Press, 2009).

    Book  Google Scholar 

  5. B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev, I. V. Volovich and E. I. Zelenov, “p-Adic mathematical physics: the first 30 years,” p-Adic Num. Ultrametr. Anal. Appl. 9, 87–121 (2017).

    Article  MathSciNet  Google Scholar 

  6. J. Falco and K.-G. Grosse-Erdmann, “Algebrability of the set of hypercyclic vectors for backward shift operators,” Adv. Math. 366, 107082 (2020).

    Article  MathSciNet  Google Scholar 

  7. N. N. Ganikhodjaev, F. M. Mukhamedov and U. A. Rozikov, “Phase transitions of the Ising model on \(\mathbb{Z}\) in the \(p\)-adic number field,” Theor. Math. Phys. 130, 425–431 (2002).

    Article  Google Scholar 

  8. R. M. Gethner and J. H. Shapiro, “Universal vectors for operators on spaces of holomorphic functions,” Proc. Amer. Math. Soc. 100 (2), 281–288 (1987).

    Article  MathSciNet  Google Scholar 

  9. G. Godefroy and J. H. Shapiro, “Operators with dense, invariant, cyclic vector manifolds,” J. Funct. Anal. 98, 229–269 (1991).

    Article  MathSciNet  Google Scholar 

  10. K.-G. Grosse-Erdmann, “Universal families and hypercyclic vectors,” Bull. Amer. Math. Soc. 36, 345–381 (1999).

    Article  MathSciNet  Google Scholar 

  11. K.-G. Grosse-Erdmann and A. Peris, Linear Chaos (Springer, Berlin, 2011).

    Book  Google Scholar 

  12. J. Bes and A. Peris, “Hereditarily hypercyclic operators,” J. Func. Anal. 167, 94–112 (1999).

    Article  MathSciNet  Google Scholar 

  13. S. Jeong, “Shift operators and two applications to \(F_q[T]\),” J. Numb. Theor. 139, 112–137 (2014).

    Article  Google Scholar 

  14. J. Kingsbery, A. Levin, A. Preygel and C. E. Silva, “Dynamics of the \(p\)-adic shift and applications,” Discr. Cont. Dyn. Sys. 30, 209–218 (2011).

    Article  MathSciNet  Google Scholar 

  15. C. Kitai, Invariant Closed Sets for Linear Operators, Thesis (University of Toronto, 1982).

    Google Scholar 

  16. A. N. Kochubei, “Non-Archimedean shift operators,” p-Adic Num. Ultrametr. Anal. Appl. 2, 260–264 (2010).

    Article  MathSciNet  Google Scholar 

  17. A. N. Kochubei, “Non-Archimedean radial calculus: Volterra operator and Laplace transform,” Integr. Equ. Oper. Theory 92, 44 (2020).

    Article  MathSciNet  Google Scholar 

  18. F. Mukhamedov, “On existence of generalized Gibbs measures for one dimensional \(p\)-adic countable state Potts model,” Proc. Steklov Inst. Math. 265, 165–176 (2009).

    Article  MathSciNet  Google Scholar 

  19. F. Mukhamedov and O. Khakimov, “Dynamics of linear operators on non-Archimedean vector spaces,” Bull. Belg. Math. Soc. 25, 85–105 (2018).

    MathSciNet  MATH  Google Scholar 

  20. F. Mukhamedov and O. Khakimov, “Chaotic behavior of the \(p\)-adic Potts-Bethe mapping,” Discr. Cont. Dyn. Sys. A 38, 231–245 (2018).

    Article  MathSciNet  Google Scholar 

  21. C. Perez-Garcia and W. H. Schikhof, Locally Convex Spaces over non-Archimedean Valued Fields (Cambridge Univ. Press, 2010).

    Book  Google Scholar 

  22. A. van Rooij, Non-Archimedean Functional Analysis (M. Dekker, New York, 1978; Cambridge Univ. Press, 2010).

    MATH  Google Scholar 

  23. H. Salas, “Hypercyclic weighted shifts,” Trans. Amer. Math. Soc. 347, 993–1004 (1995).

    Article  MathSciNet  Google Scholar 

  24. H. Salas, “Supercyclicity and weighted shifts,” Studia Math. 135, 55–74 (1999).

    Article  MathSciNet  Google Scholar 

  25. S. Shkarin, The Kitai Criterion and backward shifts, Proc. Amer. Math. Soc. 136, 1659–1670 (2006).

    Article  MathSciNet  Google Scholar 

  26. P. Schneider, Nonarchimedean Functional Analysis (Springer, 2005).

    MATH  Google Scholar 

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Funding

The present work is supported by the UAEU UPAR Grant No. G00003247 (Fund No. 31S391).

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Correspondence to Farrukh Mukhamedov, Otabek Khakimov or Abdessatar Souissi.

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Mukhamedov, F., Khakimov, O. & Souissi, A. A Few Remarks on Supercyclicity of Non-Archimedean Linear Operators on \(c_0(\mathbb N)\). P-Adic Num Ultrametr Anal Appl 14, 64–76 (2022). https://doi.org/10.1134/S2070046622010046

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  • DOI: https://doi.org/10.1134/S2070046622010046

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