Somewhere dense Cesàro orbits and rotations of Cesàro hypercyclic operators

George Costakis, and Demetris Hadjiloucas. "Somewhere dense Cesàro orbits and rotations of Cesàro hypercyclic operators." Studia Mathematica 175.3 (2006): 249-269. <http://eudml.org/doc/284909>.

@article{GeorgeCostakis2006,
abstract = {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 $\{n^\{-1\}Tⁿx\}$ 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.},
author = {George Costakis, Demetris Hadjiloucas},
journal = {Studia Mathematica},
keywords = {hypercyclic operators; Cesàro hypercyclicity; residual set},
language = {eng},
number = {3},
pages = {249-269},
title = {Somewhere dense Cesàro orbits and rotations of Cesàro hypercyclic operators},
url = {http://eudml.org/doc/284909},
volume = {175},
year = {2006},
}

TY - JOUR
AU - George Costakis
AU - Demetris Hadjiloucas
TI - Somewhere dense Cesàro orbits and rotations of Cesàro hypercyclic operators
JO - Studia Mathematica
PY - 2006
VL - 175
IS - 3
SP - 249
EP - 269
AB - 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 ${n^{-1}Tⁿx}$ 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.
LA - eng
KW - hypercyclic operators; Cesàro hypercyclicity; residual set
UR - http://eudml.org/doc/284909
ER -