Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

STRONG HYPERCYCLICITY OF BANACH SPACE OPERATORS

  • Received : 2019.12.18
  • Accepted : 2020.07.17
  • Published : 2021.01.01
Download PDF
⟨ Previous Next ⟩

Abstract

A bounded linear operator T on a separable infinite dimensional Banach space X is called strongly hypercyclic if $$X{\backslash}\{0\}{\subseteq}{\bigcup_{n=0}^{\infty}}T^n(U)$$ for all nonempty open sets U ⊆ X. We show that if T is strongly hypercyclic, then so are Tn and cT for every n ≥ 2 and each unimodular complex number c. These results are similar to the well known Ansari and León-Müller theorems for hypercyclic operators. We give some results concerning multiplication operators and weighted composition operators. We also present a result about the invariant subset problem.

Keywords

Acknowledgement

We are very grateful to the referees for their careful reading, suggestions and corrections which improved the quality of our paper.

References

  1. M. Ansari, Strong topological transitivity of some classes of operators, Bull. Belg. Math. Soc. Simon Stevin 25 (2018), no. 5, 677-685. https://projecteuclid.org/euclid.bbms/1547780428 https://doi.org/10.36045/bbms/1547780428
  2. S. I. Ansari, Hypercyclic and cyclic vectors, J. Funct. Anal. 128 (1995), no. 2, 374-383. https://doi.org/10.1006/jfan.1995.1036
  3. M. Ansari, B. Khani-Robati, and K. Hedayatian, On the density and transitivity of sets of operators, Turkish J. Math. 42 (2018), no. 1, 181-189. https://doi.org/10.3906/mat-1508-87
  4. F. Bayart and E. Matheron, Dynamics of linear operators, Cambridge Tracts in Mathematics, 179, Cambridge University Press, Cambridge, 2009. https://doi.org/10.1017/CBO9780511581113
  5. J. Bes and A. Peris, Hereditarily hypercyclic operators, J. Funct. Anal. 167 (1999), no. 1, 94-112. https://doi.org/10.1006/jfan.1999.3437
  6. G. D. Birkhoff, Surface transformations and their dynamical applications, Acta Math. 43 (1922), no. 1, 1-119. https://doi.org/10.1007/BF02401754
  7. P. S. Bourdon, Invertible weighted composition operators, Proc. Amer. Math. Soc. 142 (2014), no. 1, 289-299. https://doi.org/10.1090/S0002-9939-2013-11804-6
  8. G. Godefroy and J. H. Shapiro, Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal. 98 (1991), no. 2, 229-269. https://doi.org/10.1016/0022-1236(91)90078-J
  9. K.-G. Grosse-Erdmann and A. Peris Manguillot, Linear Chaos, Universitext, Springer, London, 2011. https://doi.org/10.1007/978-1-4471-2170-1
  10. A. Kameyama, Topological transitivity and strong transitivity, Acta Math. Univ. Comenian. (N.S.) 71 (2002), no. 2, 139-145.
  11. C. Kitai, Invariant closed sets for linear opeartors, ProQuest LLC, Ann Arbor, MI, 1982.
  12. F. Leon-Saavedra and V. Muller, Rotations of hypercyclic and supercyclic operators, Integral Equations Operator Theory 50 (2004), no. 3, 385-391. https://doi.org/10.1007/s00020-003-1299-8
  13. V. Muller and J. Vrsovsky, Orbits of linear operators tending to infinity, Rocky Mountain J. Math. 39 (2009), no. 1, 219-230. https://doi.org/10.1216/RMJ-2009-39-1-219
  14. C. J. Read, The invariant subspace problem for a class of Banach spaces. II. Hypercyclic operators, Israel J. Math. 63 (1988), no. 1, 1-40. https://doi.org/10.1007/BF02765019
  15. W. Rudin, Real and Complex Analysis, third edition, McGraw-Hill Book Co., New York, 1987.
  16. H. N. Salas, Hypercyclic weighted shifts, Trans. Amer. Math. Soc. 347 (1995), no. 3, 993-1004. https://doi.org/10.2307/2154883