A limit theorem for order preserving nonexpansive operators in L₁

We prove the following theorem: Let T be an order preserving nonexpansive operator on L₁ (μ) (or L(Formula presented.)) of a σ-finite measure, which also decreases the L_∞-norm, and let S=tI+(1−t) T for 0< t<1. Then for every f ∈ L_p (1< p<∞), the sequence Sⁿf converges weakly in L_p. (The assumptions do not imply that T is nonexpansive in L_p for any p>1, even if μ is finite.) For the proof we show that ∥Sⁿ⁺¹f−Sⁿf∥_p → 0 for every f ∈L_p, 1< p<∞, and apply to S the following theorem: Let T be order preserving and nonexpansive in L(Formula presented.), and assume that T decreases the L_∞-norm. Then for g ∈L_p (1< p<∞) Tⁿg is weakly almost convergent. If for f ∈ L_p we have Tⁿ⁺¹f−Tⁿf → 0 weakly, then Tⁿf converges weakly in L_p (1< p<∞).