Rate of growth of hypercyclic entire functions

Abstract

Let ▪ be a weighted shift operator on the space of entire functions. Suppose that | a a | → ∞ monotonically, a particular such operator being the differentiation operator Df = f′.l It is known that T possesses hypercyclic functions, that is, functions f for which ▪ is dense in ▪. In this paper a sharp result on the permissible rates of growth to T-hypercyclic entire functions is obtained, and it is shown that for every permissible growth rate there is a dense subspace of T-hypercyclic functions satisfying this growth condition. These results generalise earlier results by the author and by Armitage on the differentiation operator.