EXAMPLE OF A MEAN ERGODIC L1 OPERATOR WITH THE LINEAR RATE OF GROWTH

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 [0,1] operators T satisfying MET with lim(n ->infinity) parallel to T-n parallel to/n(1-gamma) = infinity. 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 parallel to T-n parallel to/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 ->infinity) parallel to T-n parallel to/n > 0.