Hypercyclic behavior of some non-convolution operators on $H(\mathbb{C}^N)$

TL;DR

This paper investigates the hypercyclicity of certain non-convolution operators on spaces of holomorphic functions in multiple complex variables, revealing more complex behavior than in the one-dimensional case.

Contribution

It introduces and analyzes a new class of operators combining differentiation and affine composition on $H(\,mathbb{C}^N)$, extending prior one-dimensional results.

Findings

Hypercyclicity depends on multiple parameters.

Behavior is more complex than in the one-dimensional case.

Operators generalize those studied by Aron and Markose.

Abstract

We study hypercyclicity properties of a family of non-convolution operators defined on spaces of holomorphic functions on $\mathbb{C}^N$. These operators are a composition of a differentiation operator and an affine composition operator, and are analogues of operators studied by Aron and Markose on $H(\mathbb{C})$. The hypercyclic behavior is more involved than in the one dimensional case, and depends on several parameters involved.