CONVERGENCE OF WEIGHTED AVERAGES OF MARTINGALES IN NONCOMMUTATIVE BANACH FUNCTION SPACES

Let x =(xn) n ≥1 be a martingale on a noncommutative probability space(M,τ) and(wn) n ≥1 a sequence of positive numbers such that Wn = ∑nk=1 wk →∞ as n →∞.We prove that x =(xn) n≥1 converges in E(M) if and only if(σn(x))n≥1 converges in E(M),where E(M) is a noncommutative rearrangement invariant Banach function space with the Fatou property and σ n(x) is given by k=1 If in addition,E(M) has absolutely continuous norm,then,(σ n(x)) n ≥1 converges in E(M) if and only if x =(x n) n ≥1 is uniformly integrable and its limit in measure topology x ∞∈ E(M).

Let x （xn）≥1 be a martingale on a noncommutative probability space n （M, r） and （wn）n≥1 a sequence of positive numbers such that Wn = ∑ k=1^n wk →∞ as n →∞ We prove that x = （x.）n≥1 converges in E（M） if and only if （σn（x）n≥1 converges in E（.hd）, where E（A//） is a noncommutative rearrangement invariant Banach function space with the Fatou property and σn（x） is given by σn（x） = 1/Wn ∑k=1^n wkxk, n=1, 2, .If in addition, E（Ad） has absolutely continuous norm, then, （an（x））≥1 converges in E（.M） if and only if x = （Xn）n≥1 is uniformly integrable and its limit in measure topology x∞∈ E（M）.