Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

q-FREQUENT HYPERCYCLICITY IN AN ALGEBRA OF OPERATORS

  • Heo, Jaeseong (Department of Mathematics Research Institute for Natural Sciences Hanyang University) ;
  • Kim, Eunsang (Department of Applied Mathematics College of Science and Technology Hanyang University) ;
  • Kim, Seong Wook (Department of Applied Mathematics College of Science and Technology Hanyang University)
  • Received : 2016.01.07
  • Published : 2017.03.31
Download PDF
⟨ Previous Next ⟩

Abstract

We study a notion of q-frequent hypercyclicity of linear maps between the Banach algebras consisting of operators on a separable infinite dimensional Banach space. We derive a sufficient condition for a linear map to be q-frequently hypercyclic in the strong operator topology. Some properties are investigated regarding q-frequently hypercyclic subspaces as shown in [5], [6] and [7]. Finally, we study q-frequent hypercyclicity of tensor products and direct sums of operators.

Keywords

References

  1. F. Bayart and S. Grivaux, Frequently hypercyclic operators, Trans. Amer. Math. Soc. 358 (2006), no. 11, 5083-5117. https://doi.org/10.1090/S0002-9947-06-04019-0
  2. F. Bayart and E. Matheron, Dynamics of Linear Operators, Camb. Univ. Press. 179, 2009.
  3. F. Bayart and E. Matheron, (Non)-weakly mixing operators and hypercyclicity sets, Ann. Inst. Fourier (Grenoble) 59 (2009), no. 1, 1-35. https://doi.org/10.5802/aif.2425
  4. A. Bonilla and K.-G. Grosse-Erdmann, Frequently hypercyclic operators and vectors, Ergodic Theory Dynam. Systems 27 (2007), no. 2, 383-404. https://doi.org/10.1017/S014338570600085X
  5. A. Bonilla and K.-G. Grosse-Erdmann, Frequently hypercyclic subspaces, Monatsh. Math. 168 (2012), no. 3-4, 305-320. https://doi.org/10.1007/s00605-011-0369-2
  6. K. C. Chan, Hypercyclicity of the operator algebra for a separable Hilbert space, J. Oper. Theory 42 (1999), no. 2, 231-244.
  7. K. Chan and R. Taylor, Hypercyclic subspaces of a Banach space, Integral Equations Operator Theory 41 (2001), no. 4, 381-388. https://doi.org/10.1007/BF01202099
  8. R. M. Gethner and J. H. Shapiro, Universal vectors for operators on spaces of holomor-phic functions, Proc. Amer. Math. Soc. 100 (1987), no. 2, 281-288. https://doi.org/10.1090/S0002-9939-1987-0884467-4
  9. 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
  10. K.-G. Grosse-Erdmann and A. Peris, Linear Chaos, Springer Universitext, 2011.
  11. M. Gupta and A. Mundayadan, q-frequently hypercyclic operators, Banach J. Math. Anal. 9 (2015), no. 2, 114-126. https://doi.org/10.15352/bjma/09-2-9
  12. M. Gupta and A. Mundayadan, q-frequent hypercyclicity in spaces of operators, arXiv:1407.7258[math.FA].
  13. C. Heil, A Basis Theory Primer: Expanded Edition, Birkhauser-Basel, 2011.
  14. C. Kitai, Invariant closed sets for linear operators, Ph.D. Dissertation, Univ. of Toronto, Toronoto 1982.
  15. F. Martinez-Gimenez and A. Peris, Universality and chaos for tensor products of oper-ators, J. Approx. Theory 124 (2003), no. 1, 7-24. https://doi.org/10.1016/S0021-9045(03)00118-7
  16. Q. Menet, Existence and non-existence of frequently hypercyclic subspaces for weighted shifts, Proc. Amer. Math. Soc. 143 (2015), no. 6, 2469-2477. https://doi.org/10.1090/S0002-9939-2015-12444-6