A positive doubly power bounded operator with a nonpositive inverse exists on any infinite-dimensional AL-space (Q2499026)

Let \(T\) be a positive doubly power bounded operator on a Banach space \(X\), that is, \(\sup\{\| T^n\| :n\in \mathbb{Z}\}<\infty\). It is known that \(T^{-1}\) is positive if \(T\) is defined on a finite-dimensional Banach lattice. In [Optimizatsiya 43(60), 74--80 (1988; Zbl 0696.46005)], \textit{Y.~A.\ Abramovich} proved that the positivity of \(T^{-1}\) occurs also for \(T\) being a surjective positive isometry. In [Proc.\ Am.\ Math.\ Soc.\ 132, 781--782 (2003; Zbl 1056.47026)], \textit{E.~Y.\ Emel'yanov} constructed an example of positive doubly power bounded operator with a nonpositive inverse on a special \(L_1\)-space. In the paper under review, the authors prove that such an operator exists on any infinite-dimensional \(AL\)-space.