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 . 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 In this note we construct an example of a nonpositive L¹ operator with the highest possible rate of growth, that is, .
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm124-1-2,
author = {Wojciech Kosek},
title = {Example of a mean ergodic L$^1$ operator with the linear rate of growth},
journal = {Colloquium Mathematicae},
volume = {122},
year = {2011},
pages = {15-22},
zbl = {1228.37005},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm124-1-2}
}
Wojciech Kosek. Example of a mean ergodic L¹ operator with the linear rate of growth. Colloquium Mathematicae, Tome 122 (2011) pp. 15-22. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm124-1-2/