Hypercyclicity Properties of Commutator Maps

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 ${\mathrm{\ell }}^2$. We also establish some necessary conditions which identify large classes of operators that do not induce hypercyclic commutator maps.