Supercyclic properties of extended eigenoperators of the differentiation operator on the space of entire functions

A continuous linear operator L defined on the space of entire functions H© is said to be an extended $lambda$-eigenoperator of the differentiation operator D provided DL = $lambda$LD. Here we fully characterize when an extended $lambda$-eigenoperator of D is supercyclic, it has a hypercyclic subspace or it has a supercyclic subspace.