Hypercyclic Weighted Composition Operators on ℓ^2 (Z)

Abstract

A bounded linear operator T on a separable Hilbert space H is called hypercyclic if there exists a vector x ∈ H whose orbit {T n x : n ∈ N} is dense in H . In this paper, we characterize the hypercyclicity of the weighted composition operators Cu,ϕ on ℓ 2 (Z) in terms of their weight functions and symbols. First, a necessary and sufficient condition is given for Cu,ϕ to be hypercyclic. Then, it is shown that the finite direct sums of the hypercyclic weighted composition operators are also hypercyclic. In particular, we conclude that the class of the hypercyclic weighted composition operators is weakly mixing. Finally, several examples are presented to illustrate the hypercyclicity of the weighted composition operators.

Keywords

• Hypercyclic operators,Orbit,Weighted composition operators

References

1. [1] E. Abakumov, J. Gordon, Common hypercyclic vectors for multiples of backward shift, J. Funct. Anal., 200(2), (2003), 494-504.
2. [2] F. Bayart, E. Matheron, Dynamics of linear operators, Cambridge Tract ´ s in Mathematics, Cambridge University Press, Cambridge, (2009).
3. [3] J. B`es, Dynamics of weighted composition operators, Complex Anal. Oper. Theory, 8(1), (2014), 159-176.
4. [4] J. P. B`es, A. Peris, Hereditarily hypercyclic operators, J. Funct. Anal., 167, (1999), 94-112.
5. [5] G. Costakis, A. Manoussos, J-class weighted shifts on the space of bounded sequences of complex numbers, Integr. Equ. Oper. Theory, 62(2), (2008), 149-158.
6. [6] E.A. Gallardo-Gutirrez, A. Montes-Rodrguez, The role of the spectrum in the cyclic behavior of composition operators, Mem. Amer. Math. Soc., 167(791), (2004).
7. [7] K.-G. Grosse-Erdmann, A.P. Manguillot, Linear chaos, Universitext, Springer, London, (2011).
8. [8] B. F. Madore, R. A. Mart´ınez-Avenda˜no, Subspace hypercyclicity, J. Math. Anal. Appl., 373, (2011), 502-511.

9. [9] Q. Menet, Hypercyclic subspaces and weighted shifts, Advances in Mathematics, 255, (2015), 305-337.
10. [10] S. Rolewicz. On orbits of elements, Studia Math., 32, (1969), 17-22.
11. [11] M. D. L. Rosa, C. J. Read. A hypercyclic operator whose direct sum is not hypercyclic, J. Operator Theory, 61(2), (2009), 369-380.
12. [12] H. N. Salas, Hypercyclic weighted shifts, Trans. Amer. Math. Soc., 347(3), (1995), 993-1004.
13. [13] J. H. Shapiro, Notes on dynamics of linear operators, www.math.msu.edu/shapiro, (2001).
14. [14] R. K.Singh, J. S. Manhas, Composition operators on function spaces, North-Holland Mathematics Studies, NorthHolland Publishing Co., Amsterdam, (1993).
15. [15] B. Yousefi, H. Rezaei, Hypercyclic property of weighted composition operators, Proc. Amer. Math. Soc., 135(10), (2007), 3263-3271.