# Mean Ergodic Theorems in Symmetric Spaces of Measurable Functions

M. Muratov, Yu Pashkova, B. Z. Rubshtein

Research output: Contribution to journalArticlepeer-review

## Abstract

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 $$\mathbf{L}_{1}+\mathbf{L}_{\infty}=(\mathbf{L}_{1}+\mathbf{L}_{\infty})(\Omega,\mathcal{F},\mu)$$.

Original language English 949-966 18 Lobachevskii Journal of Mathematics 42 5 https://doi.org/10.1134/S1995080221050103 Published - 1 May 2021

## Keywords

• Cesáro averages
• absolute contractions
• conservative and strictly conservative operators
• ergodic theorems
• norm convergence
• symmetric spaces

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