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

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 p F

Some results on the Orlicz space generated from a random normed module

“…Making use of this theorem, Guo and Li [12] proved the James theorem in complete RN modules; Guo, Xiao and Chen [13] established a basic strict separation theorem in random locally convex modules; Zhang and Guo [20] established a mean ergodic theorem on random reflexive RN modules, and Wu [19] proved the Bishop-Phelps theorem in complete RN modules endowed with the (ε, λ)-topology.…”

L0-Linear Modulus of a Random Linear Operator

“…In 1964, Chacon and Krengel began to study linear modulus of a linear operator and proved that there exists a unique linear modulus for a bounded linear operator [1], which plays an important role in the work of mean ergodicity for linear operators and linear operators semigroups [2][3][4][5]. Recently, the mean ergodicity for random linear operators has been investigated in [6][7][8], and its further developments should naturally include the study of 0 -linear modulus of a random linear operator on a random normed module. The purpose of this paper is to investigate the existence of the 0 -linear modulus for an a.s. bounded random linear operator on a specifical random normed module.…”

On Continuous Module Homomorphisms Between Random Locally Convex Modules

“…First, the principal relations between some basic results derived from the two kinds of topologies for random locally convex modules were studied in [8]. Then, based on these, lots of new and basic researches have recently been done in [11,12,13,17,18,19,20,21,22,23]. In this process, Guo first deeply considers the problem of applying the theory of RN modules and random locally convex modules to L 0 -conditional risk measures, for example, the Fenchel-Moreau dual representation theorem and the continuity and subdifferentiability theorems for L 0 -convex functions were pointed out in a proper form in [9], subsequently, Guo found that the study of L p -conditional risk measures can be incorporated into that of L p F (E)-conditional risk measures and further established a complete random convex analysis over random locally convex modules under the two kinds of topologies in [14].…”