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 $|\lambda|\geq 1$, the collection 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*} 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_{ \lambda,b}\circ\varphi(D): \varphi(z) \text{ is an entire function of exponential type with } \varphi(0)=0\} \end{ align*} consists entirely of hypercyclic operators.
Categories: math.FA