## Abstract

We prove the following theorem: Let T be an order preserving nonexpansive operator on L_{1} (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 S^{n}f 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 ∥S^{n+1}f−S^{n}f∥_{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<∞) T^{n}g is weakly almost convergent. If for f ∈ L_{p} we have T^{n+1}f−T^{n}f → 0 weakly, then T^{n}f converges weakly in L_{p} (1< p<∞).

Original language | English |
---|---|

Pages (from-to) | 181-191 |

Number of pages | 11 |

Journal | Israel Journal of Mathematics |

Volume | 71 |

Issue number | 2 |

DOIs | |

State | Published - 1 Jan 1990 |