Growth of hypercyclic entire functions for some non-convolution operators

Abstract

Abstract A continuous linear operator T T defined on a Fréchet space X X is said to be hypercyclic if there exists f ∈ X f\in X such that, the orbit { T n f } \left\{{T}^{n}f\right\} is dense in X X . In this article, we consider the operators introduced by Aron and Markose, defined on the space of entire functions by T λ , b f ( z ) = f ′ ( λ z + b ) {T}_{\lambda ,b}f\left(z)=f^{\prime} \left(\lambda z+b) , λ ∈ C \ { 1 } \lambda \in {\mathbb{C}}\setminus \left\{1\right\} , b ∈ C b\in {\mathbb{C}} , and we aimed to explore the rate of growth of hypercyclic vectors for T λ , b {T}_{\lambda ,b} . We discover that T λ , b {T}_{\lambda ,b} is a weighted backward shift with respect to some basis and this fact allows us to find sharp estimates of the growth of T λ , b {T}_{\lambda ,b} -hypercyclic vectors. When ∣ λ ∣ = 1 | \lambda | =1 , the T λ , b {T}_{\lambda ,b} -hypercyclic function growth is similar to the D D -hypercyclic functions ( D D is the differentiation operator), and when ∣ λ ∣ > 1 | \lambda | \gt 1 , the T λ , b {T}_{\lambda ,b} -hypercyclic functions can grow very slowly but not arbitrarily slowly. A lower bound of this growth is found in terms of the W W -Lambert function. Finally, partial results are obtained for T λ , b {T}_{\lambda ,b} -frequently hypercyclic functions.