Example of a mean ergodic $L^{1}$ operator with the linear rate of growth
Tom 124 / 2011
Colloquium Mathematicum 124 (2011), 15-22
MSC: Primary 37A30; Secondary 47A35.
DOI: 10.4064/cm124-1-2
Streszczenie
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, $\lim\sup_{n\rightarrow\infty}% {\|T^{n}\|}/{n}>0$.
