Remarks on common hypercyclic vectors

We treat the question of existence of common hypercyclic vectors for families of continuous linear operators. It is shown that for any continuous linear operator T on a complex Fréchet space X and a set Λ ⊆ R+ × C which is not of zero three-dimensional Lebesgue measure, the family {aT +bI : (a, b) ∈ Λ} has no common hypercyclic vectors. This allows to answer negatively questions raised by Godefroy and Shapiro and by Aron. We also prove a sufficient condition for a family of scalar multiples of a given operator on a complex Fréchet space to have a common hypercyclic vector. It allows to show that if D = {z ∈ C: |z| < 1} and ϕ ∈ H∞(D) is non-constant, then the family {zM ϕ: b−1 < |z| < a−1} has a common hypercyclic vector, where Mϕ : H2(D) → H2(D), Mϕf = ϕf , a = inf{|ϕ(z)|: z ∈ D} and b = sup{|ϕ(z)|: |z| ∈ D}, providing an affirmative answer to a question by Bayart and Grivaux. Finally, extending a result of Costakis and Sambarino, we prove that the family {aTb: a, b ∈ C\ {0}} has a common hypercyclic vector, where Tbf (z) = f (z − b) acts on the Fréchet space H(C) of entire functions on one complex variable. © 2009 Elsevier Inc. All rights reserved.