STUDIA MATHEMATICA 152 (3)(2002) Operators with hypercyclic Ces` aro means by Fernando Le´ on-Saavedra (C´ adiz) Abstract. An operator T on a Banach space B is said to be hypercyclic if there exists a vector x such that the orbit {T n x} n≥1 is dense in B. Hypercyclicity is a strong kind of cyclicity which requires that the linear span of the orbit is dense in B. If the arithmetic means of the orbit of x are dense in B then the operator T is said to be Ces` aro-hypercyclic. Apparently Ces` aro-hypercyclicity is a strong version of hypercyclicity. We prove that an operator is Ces` aro-hypercyclic if and only if there exists a vector x ∈B such that the orbit {n -1 T n x} n≥1 is dense in B. This allows us to characterize the unilateral and bilateral weighted shifts whose arithmetic means are hypercyclic. As a consequence we show that there are hypercyclic operators which are not Ces` aro-hypercyclic, and more surprisingly, there are non-hypercyclic operators for which the Ces` aro means of some orbit are dense. However, we show that both classes, the class of hypercyclic operators and the class of Ces` aro-hypercyclic operators, have the same norm-closure spectral characterization. 1. Introduction. Let T be a bounded linear operator on a complex Banach space B. The motivation for this work comes from some questions related to ergodic theory (see [Du], [LZ], [MZ], [Sw] for instance). The uni- form ergodic theory deals with the asymptotic behavior of the arithmetic means M n (T )= I + T + ... + T n-1 n in the operator norm (uniform) topology, as n tends to infinity. N. Dunford in 1943 (see [Du]) discussed the connections between the spectrum of T and convergence of sequences of functions Q n (T ) of T . Specifically, he obtained the following basic result in uniform ergodic theory: Theorem (Dunford). The sequence M n (T ) uniformly converges if and only if (a) lim n -1 ‖T n ‖ = 0, and 2000 Mathematics Subject Classification : 47B37, 47B38, 47B99. Key words and phrases : hypercyclic operator, hypercyclic sequences, Ces` aro means, weighted shifts, spectral characterization. Dedicado a mi tio Antonio, gracias por tantos inolvidables septiembres. [201]