Linear Factorization of Hypercyclic Functions for Differential Operators

TL;DR

This paper demonstrates that certain linear operators on entire functions have hypercyclic vectors that can be expressed as infinite products of linear polynomials, revealing a factorization property in hypercyclic dynamics.

Contribution

It establishes a linear factorization form for hypercyclic vectors of nonscalar operators commuting with differentiation on entire functions.

Findings

Hypercyclic vectors can be expressed as infinite products of linear polynomials.

Every nonscalar continuous linear operator commuting with differentiation has such hypercyclic vectors.

The hypercyclic vectors have a specific factorization structure in the space of entire functions.

Abstract

On the Fr\'{e}chet space of entire functions $H(\mathbb{C})$, we show that every nonscalar continuous linear operator $L:H(\mathbb{C})\to H(\mathbb{C})$ which commutes with differentiation has a hypercyclic vector $f(z)$ in the form of the infinite product of linear polynomials: \[ f(z) = \prod_{j=1}^\infty \, \left( 1-\frac{z}{a_j}\right), \] where each $a_j$ is a nonzero complex number.