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von Neumann’s mean ergodic theorem on complete random inner product modules

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Abstract

We first prove two forms of von Neumann’s mean ergodic theorems under the framework of complete random inner product modules. As applications, we obtain two conditional mean ergodic convergence theorems for random isometric operators which are defined on L pℱ (ℰ, H) and generated by measure-preserving transformations on Ω, where H is a Hilbert space, L p(ℰ, H) (1 ⩽ p < ∞) the Banach space of equivalence classes of H-valued p-integrable random variables defined on a probability space (Ω, ℰ, P), F a sub σ-algebra of ℰ, and L pℱ (ℰ(E,H) the complete random normed module generated by L p(ℰ, H).

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Authors and Affiliations

  1. LMIB and School of Mathematics and Systems Science, Beihang University, Beijing, 100191, China

    Xia Zhang & Tiexin Guo

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  1. Xia Zhang
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  2. Tiexin Guo
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Correspondence to Xia Zhang.

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Zhang, X., Guo, T. von Neumann’s mean ergodic theorem on complete random inner product modules. Front. Math. China 6, 965–985 (2011). https://doi.org/10.1007/s11464-011-0139-4

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  • DOI: https://doi.org/10.1007/s11464-011-0139-4

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