Operators with hypercyclic Cesaro means

Fernando León-Saavedra. "Operators with hypercyclic Cesaro means." Studia Mathematica 152.3 (2002): 201-215. <http://eudml.org/doc/285024>.

@article{FernandoLeón2002,
abstract = {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.},
author = {Fernando León-Saavedra},
journal = {Studia Mathematica},
keywords = {hypercyclic operator; hypercyclic sequences; Cesàro means; weighted shifts; spectral characterization},
language = {eng},
number = {3},
pages = {201-215},
title = {Operators with hypercyclic Cesaro means},
url = {http://eudml.org/doc/285024},
volume = {152},
year = {2002},
}

TY - JOUR
AU - Fernando León-Saavedra
TI - Operators with hypercyclic Cesaro means
JO - Studia Mathematica
PY - 2002
VL - 152
IS - 3
SP - 201
EP - 215
AB - 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.
LA - eng
KW - hypercyclic operator; hypercyclic sequences; Cesàro means; weighted shifts; spectral characterization
UR - http://eudml.org/doc/285024
ER -