Operators with hypercyclic Cesaro means

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)greater than or equal to1 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 Cesaro-hypercyclic. Apparently Cesaro-hypercyclicity is a strong version of hypercyclicity. We prove that an operator is Cesaro-hypercyclic if and only if there exists a vector x is an element of B such that the orbit {n(-1)T(n)x}(n)greater than or equal to1 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 Cesaro-hypercyclic, and more surprisingly, there are non-hypercyclic operators for which the Cesaro means of some orbit are dense. However, we show that both classes, the class of hypercyclic operators and the class of Cesaro-hypercyclic operators, have the same norm-closure spectral characterization.