Two families of hypercyclic nonconvolution 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 $C_\gamma:H(\mathbb{C})\to H(\mathbb{C})$ be the composition operator $C_\gamma f(z)=f(\lambda z+b)$, and let $D$ be the derivative operator. We extend results on the hypercyclicity of the non-convolution operators $T_{\lambda,b}=C_\gamma \circ D$ by showing that whenever $|\lambda|\geq 1$, the algebra of operators \begin{align*} \{\psi(T_{\lambda,b}): \psi(z)\in H(\mathbb{C}), \psi(0)=0 \text{ and } \psi(T_{\lambda,b}) \text{ is continuous}\} \end{align*} and the family of operators \begin{align*} \{C_\gamma\circ\varphi(D): \varphi(z) \text{ is an entire function of exponential type with } \varphi(0)=0\} \end{align*} consist entirely of hypercyclic operators (i.e., each operator has a dense orbit).