Example of a mean ergodic $L^{1}$ operator with the linear rate of growth

Wojciech Kosek¹

¹ Colorado Technical University 4435 North Chestnut Street Colorado Springs, CO 80907, U.S.A.

Colloquium Mathematicum, Tome 124 (2011) no. 1, pp. 15-22

Résumé

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 $\gamma>0,$ there are positive $L^{1}\left[ 0,1\right] $ operators $T$ satisfying MET with $\lim_{n\rightarrow\infty}\|T^{n}\|/n^{1-\gamma}=\infty.$ In the class of positive $L^{1}$ operators this is the most one can hope for in the sense that for every such operator $T$, there exists a $\gamma_{0}>0$ such that $\lim\sup\|T^{n}\|/n^{1-\gamma_{0}}=0.$ In this note we construct an example of a nonpositive $L^{1}$ operator with the highest possible rate of growth, that is, $\lim\sup_{n\rightarrow\infty}% {\|T^{n}\|}/{n}>0$.

Détail

DOI : 10.4064/cm124-1-2

Keywords: rate growth operator satisfying mean ergodic theorem met cannot faster linear recently shown kornfeld kosek colloq math every gamma there positive right operators satisfying met lim rightarrow infty gamma infty class positive operators hope sense every operator there exists gamma lim sup gamma note construct example nonpositive operator highest possible rate growth lim sup rightarrow infty