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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 $\gamma>0,$ 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, $\lim\sup_{n\rightarrow\infty}% {\|T^{n}\|}/{n}>0$.