Two Families of Hypercyclic Non-Convolution Operators

TL;DR

This paper extends the understanding of hypercyclicity in non-convolution operators on entire functions, demonstrating that certain operator families form algebras of hypercyclic operators under specific conditions.

Contribution

It introduces new classes of hypercyclic operators formed by composition and derivative operators, expanding the known families with hypercyclic properties.

Findings

Operators with $|\, ext{lambda}\,|\, extgreater=1$ form an algebra of hypercyclic operators.

Operators composed of $C_{ ext{lambda,b}}$ and entire functions of exponential type are all hypercyclic.

The results generalize previous hypercyclicity conditions for 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 $|\lambda|\geq 1$, 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.