### Abstract

We give a representation for a positive L_{p}-operator, 1 < p < ∞, in terms of a pair of positive operators (U, V), an L_{l}-operator U and an L_{∞}-operator V. This representation is obtained by an extension of the methods used in the construction of dilations of positive L_{p}-contractions to positive invertible L_{p}-isometries. A positive L_{p}-operator T and a positive L_{r}-operator H, 1 < p, r < ∞, are called associated operators if they can be represented by the same pair. If {T_{n}} is a sequence of positive L_{p}-contractions and {S_{n}} a sequence of positive L_{r}-contractions, 1 < p, r < ∞, and if S_{n} and T*_{n} are associated for each n, then we show that the sequence. S_{1}·S_{n}(T_{n}·T_{1}f) p r. converges a.e. for each nonnegative L_{p}-function f. This result includes Rota's "Alternierende Verfahren" theorem and its subsequent generalizations and covers new cases.

Original language | English (US) |
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Pages (from-to) | 249-290 |

Number of pages | 42 |

Journal | Advances in Mathematics |

Volume | 87 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1991 |

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## Cite this

*Advances in Mathematics*,

*87*(2), 249-290. https://doi.org/10.1016/0001-8708(91)90073-G