A NOTE ON FREQUENT HYPERCYCLICITY OF OPERATORS THAT $$\lambda$$-COMMUTE WITH THE DIFFERENTIATION OPERATOR

Abstract

AbstractA continuous linear operator on a Fréchet space $$\mathcal {X}$$ X is frequently hypercyclic if there exists a vector x such that for any nonempty open subset $$U\subset \mathcal {X}$$ U ⊂ X the set of $$n\in \mathbb {N}\cup \{0\}$$ n ∈ N ∪ { 0 } for which $$T^nx\in U$$ T n x ∈ U has a positive lower density. Here we determine when an operator that commutes up to a factor with the differentiation operator D, defined on the space of entire functions, is frequently hypercyclic.