Example of a mean ergodic L¹ operator with the linear rate of growth

Wojciech Kosek. "Example of a mean ergodic L¹ operator with the linear rate of growth." Colloquium Mathematicae 124.1 (2011): 15-22. <http://eudml.org/doc/283483>.

@article{WojciechKosek2011,
abstract = {The rate of growth of an operator T satisfying the mean ergodic theorem (MET) cannot be faster than linear. It was recently shown (Kornfeld-Kosek, Colloq. Math. 98 (2003)) that for every γ > 0, there are positive L¹[0,1] operators T satisfying MET with $lim_\{n→ ∞\}||Tⁿ||/n^\{1-γ\} = ∞$. In the class of positive L¹ operators this is the most one can hope for in the sense that for every such operator T, there exists a γ₀ > 0 such that $lim sup||Tⁿ||/n^\{1-γ₀\} = 0.$ In this note we construct an example of a nonpositive L¹ operator with the highest possible rate of growth, that is, $lim sup_\{n → ∞\}||Tⁿ||/n > 0$.},
author = {Wojciech Kosek},
journal = {Colloquium Mathematicae},
keywords = {mean ergodic operator; rate of growth},
language = {eng},
number = {1},
pages = {15-22},
title = {Example of a mean ergodic L¹ operator with the linear rate of growth},
url = {http://eudml.org/doc/283483},
volume = {124},
year = {2011},
}

TY - JOUR
AU - Wojciech Kosek
TI - Example of a mean ergodic L¹ operator with the linear rate of growth
JO - Colloquium Mathematicae
PY - 2011
VL - 124
IS - 1
SP - 15
EP - 22
AB - The rate of growth of an operator T satisfying the mean ergodic theorem (MET) cannot be faster than linear. It was recently shown (Kornfeld-Kosek, Colloq. Math. 98 (2003)) that for every γ > 0, there are positive L¹[0,1] operators T satisfying MET with $lim_{n→ ∞}||Tⁿ||/n^{1-γ} = ∞$. In the class of positive L¹ operators this is the most one can hope for in the sense that for every such operator T, there exists a γ₀ > 0 such that $lim sup||Tⁿ||/n^{1-γ₀} = 0.$ In this note we construct an example of a nonpositive L¹ operator with the highest possible rate of growth, that is, $lim sup_{n → ∞}||Tⁿ||/n > 0$.
LA - eng
KW - mean ergodic operator; rate of growth
UR - http://eudml.org/doc/283483
ER -