Mean Ergodic Theorems in Symmetric Spaces of Measurable Functions

Abstract: Let $$\mathbf{E}=\mathbf{E}(\Omega,\mathcal{F},\mu)$$ be a symmetric Banach space of measurable functions on a measure space $$(\Omega,\mathcal{F},\mu)$$. We prove a version of Mean (Statistical) Ergodic Theorem for Cesáro averages $$A_{n,T}f=1/n\sum_{k=1}^{n}T^{k-1}f$$, $$f\in\mathbf{E}$$, while operators on $$\mathbf{E}$$ are induced by positive absolute contraction in $$\mathbf{L}_{1}+\mathbf{L}_{\infty}=(\mathbf{L}_{1}+\mathbf{L}_{\infty})(\Omega,\mathcal{F},\mu)$$.