Random δ-nearsurjective ε-isometries on random normed modules

Release time:2024-10-25

Journal:J. Nonlinear Convex Anal.

Key Words:approximation Random normed modules random operators random δ–nearsurjective ε–isometries sample-continuous sample-linear sample-nonexpansive

Abstract:Let (Ω, F, P) be a probability space, R the scalar field of real numbers, L0(F,R) the equivalence classes of iZ-valued F-measurable random variables on Q, (E, || . ||) and (F, || . ||) two complete random normed modules over R with base (Ω, F, P). The main theorem of this paper is the following approximation result for random δ–nearsurjective ε–isometries between random normed modules: if f : E → F is a stable random δ–nearsurjective ε–isometry with f(0) = 0, where s, ε, λ ∈ L0(F, R) and ε, λ ≥ 0, then there exists a surjective L0-linear random isometry U between E and F such that ||f(x) - U(x)|| ≤ 4ε for all x ∈ E. Furthermore, making use of the above result and the relations between random normed modules and classical normed spaces, we give the approximation result for sample-continuous random operators: let (X, || - ||) and (Y, || - ||) be two real separable Banach spaces, ε0 and λ0 two nonnegative random variables and f: Ω × X → Y a random operator such that f(ω, .) : X → Y is a continuous εS0(u;)-nearsurjective λ0(ω)-isometry and f(ω, 0) = 0 for any w ∈ Ω, then there exist a sample-linear and almost everywhere (briefly, a.e.) isometric random operator U: Q × X → Y and Ω0 ∈ F with P(Ω0) = 1 such that ||f(ω,x) - U(ω,x)|| ≤ 4λ0(ω), V(ω,x) ∈ Ω0 × X. It is the first time that sample-linear and a.e. isometric random operators are used to approximate sample-continuous nonlinear random operators.

Co-author:Tiexin Guo

First Author:Yachao Wang

Indexed by:Journal paper

Discipline:Natural Science

First-Level Discipline:Mathematics

Document Type:J

Volume:24

Issue:3

Page Number:535-551

ISSN No.:1880-5221

Translation or Not:no

Date of Publication:2023-10-26

Included Journals:SCI

Links to published journals:http://yokohamapublishers.jp/online2/opjnca/vol24/p535.html