Operators with hypercyclic Cesaro means

Fernando León-Saavedra
2002 Studia Mathematica  
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 Cesàro-hypercyclic. Apparently Cesàro-hypercyclicity is a strong version of hypercyclicity. We prove that an operator is Cesàro-hypercyclic if and only if there
more » ... 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 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. 2000 Mathematics Subject Classification: 47B37, 47B38, 47B99.
doi:10.4064/sm152-3-1 fatcat:sulmlgli6zgbzfph276hwiapee