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 Frechet space X and a set Lambda subset of R(+) x C which is not of zero three-dimensional Lebesgue measure, the family {aT + bl: (a, b) is an element of Lambda} 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 Frechet space to have a common hypercyclic vector. It allows to show that if D = {z is an element of C: vertical bar z vertical bar < 1} and phi is an element of H(infinity)(D) is non-constant, then the family {zM(phi)(star): b(-1) < vertical bar z vertical bar < a(-1)} has a common hypercyclic vector, where M(phi) : H(2)(D) -> H(2)(D), M(phi)f = phi f, a = inf{vertical bar phi(z)vertical bar: z is an element of D} and b = sup{vertical bar phi(z)vertical bar: vertical bar z vertical bar is an element of 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 {aT(b): a, b is an element of C\{0}} has a common hypercyclic vector, where T(b)f(z) = f(z - b) acts on the Frechet space H(C) of entire functions on one complex variable. (C) 2009 Elsevier Inc. All rights reserved.