Let T be a continuous linear operator acting on a Banach space X. We examine whether certain fundamental results for hypercyclic operators are still valid in the Cesàro hypercyclicity setting. In particular, in connection with the somewhere dense orbit theorem of Bourdon and Feldman, we show that if for some vector x ∈ X the set Tx,T²/2 x,T³/3 x, ... is somewhere dense then for every 0 < ε < 1 the set (0,ε)Tx,T²/2 x,T³/3 x,... is dense in X. Inspired by a result of Feldman, we also prove that if the sequence is d-dense then the operator T is Cesàro hypercyclic. Finally, following the work of León-Saavedra and Müller, we consider rotations of Cesàro hypercyclic operators and we establish that in certain cases, for any λ with |λ | = 1, T and λT share the same sets of Cesàro hypercyclic vectors.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm175-3-4,
author = {George Costakis and Demetris Hadjiloucas},
title = {Somewhere dense Ces\`aro orbits and rotations of Ces\`aro hypercyclic operators},
journal = {Studia Mathematica},
volume = {173},
year = {2006},
pages = {249-269},
zbl = {1161.47006},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm175-3-4}
}
George Costakis; Demetris Hadjiloucas. Somewhere dense Cesàro orbits and rotations of Cesàro hypercyclic operators. Studia Mathematica, Tome 173 (2006) pp. 249-269. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm175-3-4/