Existence of common hypercyclic subspaces for the derivative operator and the translation operators

TL;DR

This paper proves that certain operators on entire functions, specifically derivatives and translations, share a common hypercyclic subspace where all non-zero vectors have dense orbits, revealing deep dynamical properties.

Contribution

It establishes the existence of a common hypercyclic subspace for both derivative and translation operators on entire functions, a novel result in operator dynamics.

Findings

Non-zero multiples of derivative and translation operators share a hypercyclic subspace.

Every non-zero vector in this subspace has a dense orbit under both operators.

The result advances understanding of operator dynamics on function spaces.

Abstract

We show that the non-zero multiples of the derivative operator and the non-zero multiples of non-trivial translation operators on the space of entire functions share a common hypercyclic subspace, i.e. a closed infinite-dimensional subspace in which each non-zero vector has a dense orbit for each of these operators.