Abstract:In this paper we study an operator $T$ on a Banach space $E$ which is $M$-hypercyclic for some subspace $M$ of $E$. We give a sufficient condition for such an operator to be $M$-hypercyclic and use it to answer negatively two questions asked by Madore and Martínez-Avendaño. We also give a sufficient condition for $T$ to be $M$-hypercyclic for all finite co-dimensional subspaces $M$ in $E$.
- Frédéric Bayart and Étienne Matheron, Dynamics of linear operators, Cambridge Tracts in Mathematics, vol. 179, Cambridge University Press, Cambridge, 2009. MR 2533318, DOI 10.1017/CBO9780511581113
- Gilles Godefroy and Joel H. Shapiro, Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal. 98 (1991), no. 2, 229–269. MR 1111569, DOI 10.1016/0022-1236(91)90078-J
- Karl-Goswin Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 3, 345–381. MR 1685272, DOI 10.1090/S0273-0979-99-00788-0
- B. F. Madore and R. A. Martínez-Avendaño. Subspace hypercyclicity, J. Math. Anal. Appl. 373(2): 502-511, 2011.
- Can Minh Le
- Affiliation: Department of Mathematics and Sciences, Kent State University, Kent, Ohio 44242
- Email: firstname.lastname@example.org
- Received by editor(s): May 4, 2010
- Received by editor(s) in revised form: July 28, 2010
- Published electronically: January 7, 2011
- Additional Notes: The author would like to express his thanks to Professor Richard M. Aron for his invaluable advice.
- Communicated by: Nigel J. Kalton
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
- Journal: Proc. Amer. Math. Soc. 139 (2011), 2847-2852
- MSC (2010): Primary 47A16; Secondary 47B37, 37B99
- DOI: https://doi.org/10.1090/S0002-9939-2011-10754-8
- MathSciNet review: 2801626