Two Families of Hypercyclic Non-Convolution Operators

Abstract

Let $H(\mathbb{C})$ be the set of all entire functions endowed with the topology of uniform convergence on compact sets. Let $\lambda ,b\in \mathbb{C}$, let $C_{\lambda ,b}:H(\mathbb{C})\to H(\mathbb{C})$ be the composition operator $C_{\lambda ,b}f(z)=f(\lambda z+b)$, and let $D$ be the derivative operator. We extend results on the hypercyclicity of the non-convolution operators $T_{\lambda ,b}=C_{\lambda ,b}\circ D$ by showing that whenever $|\lambda |\ge 1$, the collection of operators \begin{align*} \{ψ(T_{λ,b}): ψ(z)\in H(\mathbb{C}), ψ(0)=0 \text{ and } ψ(T_{λ,b}) \text{ is continuous}\} \end{align*} forms an algebra under the usual addition and multiplication of operators which consists entirely of hypercyclic operators (i.e., each operator has a dense orbit). We also show that the collection of operators \begin{align*} \{C_{λ,b}\circφ(D): φ(z) \text{ is an entire function of exponential type with } φ(0)=0\} \end{align*} consists entirely of hypercyclic operators.