Operators with hypercyclic Cesaro means
Volume 152 / 2002
Abstract
An operator $T$ on a Banach space ${\cal B}$ is said to be hypercyclic if there exists a vector $x$ such that the orbit $\{ T^nx\} _{n\geq 1}$ is dense in ${\cal B}$. Hypercyclicity is a strong kind of cyclicity which requires that the linear span of the orbit is dense in ${\cal B}$. If the arithmetic means of the orbit of $x$ are dense in ${\cal B}$ then the operator $T$ is said to be Cesàro-hypercyclic. Apparently Cesàro-hypercyclicity is a strong version of hypercyclicity. We prove that an operator is Cesàro-hypercyclic if and only if there exists a vector $x\in {\cal B}$ such that the orbit $\{ n^{-1}T^nx\} _{n\geq 1}$ is dense in ${\cal B}$. 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 Cesàro-hypercyclic, and more surprisingly, there are non-hypercyclic operators for which the Cesàro means of some orbit are dense. However, we show that both classes, the class of hypercyclic operators and the class of Cesàro-hypercyclic operators, have the same norm-closure spectral characterization.
