An operator T on a Banach space ℬ is said to be hypercyclic if there exists a vector x such that the orbit 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 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm152-3-1,
author = {Fernando Le\'on-Saavedra},
title = {Operators with hypercyclic Cesaro means},
journal = {Studia Mathematica},
volume = {151},
year = {2002},
pages = {201-215},
zbl = {1041.47004},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm152-3-1}
}
Fernando León-Saavedra. Operators with hypercyclic Cesaro means. Studia Mathematica, Tome 151 (2002) pp. 201-215. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm152-3-1/