PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 136, Number 2, February 2008, Pages 519–528 S 0002-9939(07)09036-3 Article electronically published on October 24, 2007 HYPERCYCLIC AND TOPOLOGICALLY MIXING COSINE FUNCTIONS ON BANACH SPACES ANTONIO BONILLA AND PEDRO J. MIANA (Communicated by N. Tomczak-Jaegermann) Abstract. Our first aim in this paper is to give sufficient conditions for the hypercyclicity and topological mixing of a strongly continuous cosine function. We apply these results to study the cosine function associated to translation groups. We also prove that every separable infinite dimensional complex Ba- nach space admits a topologically mixing uniformly continuous cosine family. Introduction A bounded linear operator T ∈B(X) on a separable complex Banach space X is said to be hypercyclic if there exists an x ∈ X such that {T n x} n∈N is dense in X. In 1969, Rolewicz [12] gave the first example of a hypercyclic operator on a Banach space. He showed that if B is the backward shift on l 2 (N), then λB is hypercyclic if and only if |λ| > 1. He also wondered if for every separable infinite dimensional Banach space there exists a hypercyclic operator. This question was independently answered in the affirmative by Ansari [1] and Bernal-Gonz´ alez [5]. Bonet and Peris [7] generalized the result for Fr´ echet spaces. It is well known that T ∈B(X) is hypercyclic if and only if for any pair of nonvoid open sets U, V ⊂ X there exists some positive integer n 0 such that (∗) T n 0 U ∩ V = ∅. It is said that an operator is topologically mixing if the condition (∗) holds for every n large enough. A one-parameter family {T (t)} t≥0 ⊂B(X) of bounded linear operators is a one- parameter semigroup of operators in B(X) if it verifies the following two conditions: (i) T (0) = I , (ii) T (t)T (s)= T (t + s) for all t, s ≥ 0. Received by the editors July 17, 2006. 2000 Mathematics Subject Classification. Primary 47D09, 47A16. Key words and phrases. Hypercyclic operators, topologically mixing operators, cosine func- tions, translation groups. The first author is supported by MEC and FEDER MTM2005-07347 and MEC (Accion special) MTM2006-26627-E. The second author is supported by Project MTM2004-03036, DGI-FEDER, of the MCYT, Spain, and Project E-64, D. G. Arag´on, Spain. c 2007 American Mathematical Society 519