Positive $L^1$ operators associated with nonsingular mappings and an example of E. Hille
Volume 98 / 2003
Abstract
E. Hille [Hi1] gave an example of an operator in $L^1[0,1]$ satisfying the mean ergodic theorem (MET) and such that $\mathop {\rm sup}_n \| T^n\| =\infty $ (actually, $\| T^n\| \sim n^{{1/4}})$. This was the first example of a non-power bounded mean ergodic $L^1$ operator. In this note, the possible rates of growth (in $n$) of the norms of $T^n$ for such operators are studied. We show that, for every $\gamma >0$, there are positive $L^1$ operators $T$ satisfying the MET with $\mathop {\rm lim}_{n\to \infty } {\| T^n\| /n^{1-\gamma }}=\infty $. In the class of positive operators these examples are the best possible in the sense that for every such operator $T$ there exists a $\gamma _0>0$ such that $\mathop {\rm lim}\mathop {\rm sup}_{n \to \infty } {\| T^n\| /n^{1-\gamma _0}}=0$.
A class of numerical sequences $\{ \alpha _n\} $, intimately related to the problem of the growth of norms, is introduced, and it is shown that for every sequence $\{ \alpha _n\} $ in this class one can get $\| T^n\| \geq \alpha _n$ $(n=1,2,\dots)$ for some $T$. Our examples can be realized in a class of positive $L^1$ operators associated with piecewise linear mappings of $[0,1]$.
