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

Research Article

DOI: 10.1007/s11464-011-0139-4

Cite this article as:
Zhang, X. & Guo, T. Front. Math. China (2011) 6: 965. doi:10.1007/s11464-011-0139-4

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 Lp(ℰ, H) and generated by measure-preserving transformations on Ω, where H is a Hilbert space, Lp(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 Lp(ℰ(E,H) the complete random normed module generated by Lp(ℰ, H).

Keywords

Random inner product module random normed module random unitary operator random contraction operator von Neumann’s mean ergodic theorem 

MSC

47A35 46H25 46C50 47B80 60G57 

Copyright information

© Higher Education Press and Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.LMIB and School of Mathematics and Systems ScienceBeihang UniversityBeijingChina

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