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• # Artykuł - szczegóły

## Studia Mathematica

2002 | 152 | 3 | 201-215

## Operators with hypercyclic Cesaro means

EN

### Abstrakty

EN
An operator T on a Banach space ℬ is said to be hypercyclic if there exists a vector x such that the orbit ${Tⁿx}_{n≥1}$ 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 exists a vector x ∈ ℬ such that the orbit ${n^{-1}Tⁿx}_{n≥1}$ 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.

201-215

wydano
2002

### Twórcy

• Departamento de Matemáticas, Escuela Superior de Ingeniería, Universidad de Cádiz, C/Sacramento 82, 11003 Cádiz, Spain