Positive $L^1$ operators associated with nonsingular mappings and an example of E. Hille

Isaac Kornfeld¹ ; Wojciech Kosek²

¹ Department of Mathematics North Dakota State University Fargo, ND 58105, U.S.A.
² Department of Mathematics and Statistics South Dakota State University Brookings, SD 57007, U.S.A.

Colloquium Mathematicum, Tome 98 (2003) no. 1, pp. 63-77

Résumé

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 $[0,1]$.

Détail

DOI : 10.4064/cm98-1-5

Keywords: hille gave example operator satisfying mean ergodic theorem met mathop sup infty actually sim first example non power bounded mean ergodic operator note possible rates growth norms operators studied every gamma there positive operators satisfying met mathop lim infty gamma infty class positive operators these examples best possible sense every operator there exists gamma mathop lim mathop sup infty gamma class numerical sequences alpha intimately related problem growth norms introduced shown every sequence alpha class get geq alpha dots examples realized class positive operators associated piecewise linear mappings