Hypercyclicity Properties of Commutator Maps

TL;DR

This paper studies the hypercyclic behavior of commutator maps on operator ideals, showing that certain scalar multiples of the backward shift do not produce hypercyclic maps and identifying conditions for non-hypercyclicity.

Contribution

It provides new results on when commutator maps are not hypercyclic, especially for scalar multiples of the backward shift operator.

Findings

Scalar multiples of the backward shift do not induce hypercyclic commutator maps.

Necessary conditions for non-hypercyclicity of commutator maps are established.

Large classes of operators are identified that do not produce hypercyclic commutator maps.

Abstract

We investigate the hypercyclic properties of commutator maps acting on separable ideals of operators. As the main result we prove the commutator map induced by scalar multiples of the backward shift operator fails to be hypercyclic on the space of compact operators on $\ell^2$. We also establish some necessary conditions which identify large classes of operators that do not induce hypercyclic commutator maps.