Mean ergodic theorems in symmetric spaces of measurable functions
Let \((\Omega,\mathcal{F},\mu)\) be a finite or \(\sigma\)-finite non-atomic measure space and let \(E=E(\Omega,\mathcal{F},\mu)\) be the symmetric space of measurable functions on \((\Omega,\mathcal{F},\mu)\). Let \(T\) be a positive linear operator on \(L^1(\Omega,\mathcal{F},\mu)+L^\infty(\Omega,\mathcal{F},\mu)\) such that \(T|_{L^1(\Omega,\mathcal{F},\mu)}\) and \(T|_{L^\infty(\Omega,\mathcal{F},\mu)}\) are contractions on \(L^1(\Omega,\mathcal{F},\mu)\) and \(L^\infty(\Omega,\mathcal{F},\mu)\), respectively. The minimal part \(E^0=E^0(\Omega,\mathcal{F},\mu)\) of \(E\) is defined as the closure in \(E\) of \(L^1(\Omega,\mathcal{F},\mu)\cap L^\infty(\Omega,\mathcal{F},\mu)\). The main result of the paper is the following version of the mean ergodic theorem for positive contractions in symmetric spaces. If \(L^1(\Omega,\mathcal{F},\mu)\not\subseteq E\) and \(E\not\subseteq L^\infty(\Omega,\mathcal{F},\mu)\), then for every \(f\in E^0\) the sequence \[ \frac{1}{n}\sum_{k=1}^n T^{k-1}f \] is norm convergent in \(E\) as \(n\) goes to infinity. Pointwise and stochastic ergodic theorems are also discussed.
scientific article; zbMATH DE number 7357844
scientific article