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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 $li{m}_{n\to \infty }\left|\right|Tⁿ\left|\right|/n^{1-\gamma }=\infty$. 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 $limsup\left|\right|Tⁿ\left|\right|/n^{1-\gamma ₀}=0.$ In this note we construct an example of a nonpositive L¹ operator with the highest possible...

E. Hille [Hi1] gave an example of an operator in L¹[0,1] satisfying the mean ergodic theorem (MET) and such that supₙ||Tⁿ|| = ∞ (actually, $\left|\right|Tⁿ\left|\right|\sim n^{1/4}$). This was the first example of a non-power bounded mean ergodic L¹ operator. In this note, the possible rates of growth (in n) of the norms of Tⁿ for such operators are studied. We show that, for every γ > 0, there are positive L¹ operators T satisfying the MET with $li{m}_{n\to \infty }\left|\right|Tⁿ\left|\right|/n^{1-\gamma }=\infty .IntheclassofpositiveoperatorstheseexamplesarethebestpossibleinthesensethatforeverysuchoperatorTthereexistsa\gamma ₀>0suchthat$lim supn→ ∞ ||Tⁿ||/n1-γ₀ = 0$.$A class of numerical sequences αₙ, intimately related to the...