Mean ergodicity on Banach lattices and Banach spaces
Let \(E\) be a Banach space. The linear operator \(T:E\rightarrow E\) is called power-bounded if \(\sup\{||T^n||:n\in \mathbb N\}<\infty\). \(T\) is mean ergodic whenever the sequence \((\frac{1}{n+1}\sum_{k=0}^nT^kx)\) converges in \(E\) for every \(x\in E\). Now let \(E\) be a Banach lattice. \(T\) is called power-order-bounded if, for every \(x\in E^+\), there exists \(z\in E^+\) such that \(T^n([-x,x])\subseteqq [-z,z]\) for all \(n\in \mathbb N\). A contractive operator \(T\) possesses a quasi order bounded attractor if there exist \(z>0\) and \(\eta\in [0,1]\) such that \(\lim_{n\to\infty}\inf\{\|T^nx-y\|: y\in [-z,z]+B(0,\eta)\}=0\) where \(B(0,\eta)\) denotes the closed ball around \(0\) with radius \(\eta\). In the paper under review the following facts are established: (i) The Banach lattice \(E\) has order continuous norm iff every power-order-bounded operator \(T\) is mean ergodic. (Theorem 1.1) (ii) A countably order complete Banach lattice is a KB-space iff every positive operator which possesses a quasi order bounded attractor is mean ergodic. (Theorem 2.2) (iii) A Banach space does not contain \(c_0\) if every power bounded Fredholm operator with index \(0\) is mean ergodic. (Theorem 3.1.).
mean ergodic and power bounded operators
scientific article; zbMATH DE number 1295493