PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 137, Number 11, November 2009, Pages 3787–3795 S 0002-9939(09)09943-2 Article electronically published on June 5, 2009 COMMON HYPERCYCLIC FUNCTIONS FOR MULTIPLES OF CONVOLUTION AND NON-CONVOLUTION OPERATORS LUIS BERNAL-GONZ ´ ALEZ (Communicated by Nigel J. Kalton) Dedicated to the memory of Professor Antonio Aizpuru, who died in March 2008 Abstract. We prove the existence of a residual set of entire functions, all of whose members are hypercyclic for every non-zero scalar multiple of T , where T is the differential operator associated to an entire function of order less than 1/2. The same result holds if T is a finite-order linear differential operator with non-constant coefficients. 1. Introduction In this paper, we are concerned with the existence of vectors having dense orbit with respect to each member of a non-denumerable family of operators. Specifically, we deal with the problem of the existence of entire functions that are simultaneously hypercyclic with respect to all non-zero scalar multiples either of a convolution operator or of a linear differential operator with non-constant coefficients. Precise definitions are given below, in this section and in the next one. See [22] and [23] for excellent surveys about hypercyclicity. Assume that X is a Hausdorff topological vector space and that T : X → X is a (linear, continuous) operator in it. Then T is said to be hypercyclic provided that there is a vector x ∈ X, called hypercyclic for T , whose orbit {T k x : k =0, 1, 2,... } under T is dense in X. By HC (T ) we denote the subset of all hypercyclic vectors for T . Note that the separability of X is a necessary condition in order that HC (T ) = ∅. It is easy to see that HC (T ) is dense in X if T is hypercyclic. If in addition, X is an F-space (that is, X is a completely metrizable topological vector space), then HC (T ) is a dense G δ subset of X; in particular, HC (T ) is residual in X. Denote N = {1, 2,... }. By Baire’s category theorem, if {T n : n ∈ N} is a denumerable family of hypercyclic vectors on an F-space X, then  ∞ n=1 HC (T n ) is still residual (hence non-empty) in X. But Baire’s theorem is no longer at our disposal when we Received by the editors July 7, 2008, and, in revised form, February 23, 2009. 2000 Mathematics Subject Classification. Primary 47A16; Secondary 30E10, 47B33. Key words and phrases. Hypercyclic operators, common hypercyclic vectors, entire functions, linear differential operators, Borel transform. The author has been partially supported by the Plan Andaluz de Investigaci´on de la Junta de Andaluc´ ıa FQM-127, by MEC Grant MTM2006-13997-C02-01 and by MEC Acci´on Especial MTM2006-26627-E. c 2009 American Mathematical Society Reverts to public domain 28 years from publication 3787 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use