TITLE:
Uniform Convergence of Translation Operators
AUTHORS:
Nikolaos Tsirivas
KEYWORDS:
Hypercyclic Operator, Common Hypercyclic Vectors, Translation Operator
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.12 No.12,
December
12,
2022
ABSTRACT: We denote N, R, Cthe sets of natural, real and complex numbers respectively. Let (λn), n ∈ N be an unbounded sequence of complex numbers. Costakis has proved the following result. There exists an entire function f with the following property: for every x, y ∈ R with 0 , every θ ∈(0,1) and every a ∈ C there is a subsequence of natural numbers (mn), n ∈ N such that, for every compact subset L ⊆ C , In the present paper we show that the constant function a cannot be replaced by any non-constant entire function G. This is so even if one demands the convergence in (*) only for a single radius r and a single positive number θ. This result is related with the problem of existence of common universal vectors for an uncountable family of sequences of translation operators.