Hypercyclic Weighted Composition Operators on ℓ^2 (Z)

Year 2017, Volume: 14 Issue: 2, - , 01.11.2017

Mohammad Reza Azimi

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

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