Uniform Convergence of Translation Operators

Abstract

Let θ be a fixed positive number, θ ∈ (0, 1) and let λ = (λn)n∈N be a fixed sequence of non-zero complex numbers, so that λn→∞. We shall apply the functions gn : [0, θ]× C→C, defined as gn((t, z)) = z + λne for each (t, z) ∈ [0, θ]× C. We shall consider the space C([0, θ]× C) of continuous functions on [0, θ]× C, as endowed with the topology of uniform convergence on compacta and let ρ be the usual metric in C([0, θ]×C). For an entire function f ∈ H(C) we shall denote that f̄ : [0, θ]× C→C, f̄((t, z)) = f(z) for every (t, z) ∈ [0, θ]× C. We will prove that the equation: lim n→+∞ ρ((x ◦ gyn , f̄)) = 0 does not have any solution (x, yn) where x ∈ H(C) and yn is an strictly increasing subsequence of natural numbers and f ∈ H(C) is a given non-constant entire function. When f is a constant entire function, then the above equation has infinitely several solutions, according to a result provided by G. Costakis.