• Received : 2020.04.03
  • Accepted : 2020.08.18
  • Published : 2020.10.31


In this paper, we introduce and investigate multi subspace-hypercyclic operators and prove that multi-hypercyclic operators are multi subspace-hypercyclic. We show that if T is M-hypercyclic or multi M-hypercyclic, then Tn is multi M-hypercyclic for any natural number n and by using this result, make some examples of multi subspace-hypercyclic operators. We prove that multi M-hypercyclic operators have somewhere dense orbits in M. We show that analytic Toeplitz operators can not be multi subspace-hypercyclic. Also, we state a sufficient condition for coanalytic Toeplitz operators to be multi subspace-hypercyclic.



  1. N. Bamerni, V. Kadets, and A. Kilicman, Hypercyclic operators are subspace hypercyclic, J. Math. Anal. Appl. 435 (2016), no. 2, 1812-1815.
  2. N. Bamerni and A. Kilicman, On subspace-diskcyclicity, Arab J. Math. Sci. 23 (2017), no. 2, 133-140.
  3. P. S. Bourdon and N. S. Feldman, Somewhere dense orbits are everywhere dense, Indiana Univ. Math. J. 52 (2003), no. 3, 811-819.
  4. G. Godefroy and J. H. Shapiro, Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal. 98 (1991), no. 2, 229-269.
  5. K.-G. Grosse-Erdmann and A. Peris Manguillot, Linear Chaos, Universitext, Springer, London, 2011.
  6. D. A. Herrero, Hypercyclic operators and chaos, J. Operator Theory 28 (1992), no. 1, 93-103.
  7. R. R. Jimenez-Munguia, R. A. Martinez-Avendano, and A. Peris, Some questions about subspace-hypercyclic operators, J. Math. Anal. Appl. 408 (2013), no. 1, 209-212.
  8. C. M. Le, On subspace-hypercyclic operators, Proc. Amer. Math. Soc. 139 (2011), no. 8, 2847-2852.
  9. B. F. Madore and R. A. Martinez-Avendano, Subspace hypercyclicity, J. Math. Anal. Appl. 373 (2011), no. 2, 502-511.
  10. R. A. Martinez-Avendano and P. Rosenthal, An Introduction to Operators on the Hardy-Hilbert Space, Graduate Texts in Mathematics, 237, Springer, New York, 2007.
  11. R. A. Martinez-Avendano and O. Zatarain-Vera, Subspace hypercyclicity for Toeplitz operators, J. Math. Anal. Appl. 422 (2015), no. 1, 772-775.
  12. V. G. Miller, Remarks on finitely hypercyclic and finitely supercyclic operators, Integral Equations Operator Theory 29 (1997), no. 1, 110-115.
  13. M. Moosapoor, Common subspace-hypercyclic vectors, Int. J. Pure Apll. Math. 118 (2018), no. 4, 865-870.
  14. A. Peris, Multi-hypercyclic operators are hypercyclic, Math. Z. 236 (2001), no. 4, 779-786.