HYPERCYCLICITY FOR TRANSLATIONS THROUGH RUNGE'S THEOREM

Abstract

In this paper, we first adapt Runge’s Theorem to work on certain domains in any complex Banach space. Then, using this result, we extend Birkhoff’s Theorem on the hypercyclicity of translations on H(C) and Costakis’ and Sambarino’s result on the existence of common hypercyclic functions for uncountable families of translations on H(C) to subspaces of Hb(E) (in some cases all of Hb(E)), E being in a large class of Banach spaces. A continuous linear operator T : E → E on a Frechet space E is called hypercyclic if there is a vector x ∈ E such that its orbit under T , given by O(x; T ) = {x, Tx, T x, . . .}, is dense in E (in this case the vector x is called a hypercyclic vector for the operator T ). The first known example of hypercyclic operator comes through Birkhoff’s Theorem [3], in 1929. Birkhoff showed that there is a function f in the space H(C) of entire functions on C and a sequence (an) of positive numbers such that the translates {f(z), f(z + a1), f(z + a2), . . .} are dense in H(C) (considering the compact-open topology). Actually this doesn’t match with the above definition, but in Birkhoff’s proof the a1, a2, . . . can be chosen as multiples of any real positive number a. So we have that, for any a > 0, the translation by a, Ta : H(C) → H(C), given by Ta(f)(z) = f(z+a) , is a hypercyclic operator. It’s also easy to see that if f ∈ H(C) is hypercyclic for the translation Ta, a > 0, then g(z) = f(e−iθz) is hypercyclic for Taeiθ , for each θ ∈ [0, 2π]. So, for every b 6= 0 in C, the translation Tb : H(C) → H(C) is a hypercyclic operator. In the study of hypercyclicity for translations on H(C) one particular tool has been shown to be very useful, namely: Theorem 1 (Runge). If f is holomorphic in a neighborhood of a compact set K ⊂ C and C\K is connected, then f can be uniformly approximated on K by polynomials. (see [4], p.85) Runge’s Theorem becomes a natural tool when we deal with hypercyclicity for translations not only because we are considering the compact-open topology Received February 17, 2006; Revised July 27, 2006. 2000 Mathematics Subject Classification. Primary 47A16, Secondary 46G20.