Representation of positive operators and alternating sequences

M. A. Akcoglu, J. R. Baxter, W. M.F. Lee

Research output: Contribution to journalArticle

3 Scopus citations

Abstract

We give a representation for a positive Lp-operator, 1 < p < ∞, in terms of a pair of positive operators (U, V), an Ll-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 Lp-contractions to positive invertible Lp-isometries. A positive Lp-operator T and a positive Lr-operator H, 1 < p, r < ∞, are called associated operators if they can be represented by the same pair. If {Tn} is a sequence of positive Lp-contractions and {Sn} a sequence of positive Lr-contractions, 1 < p, r < ∞, and if Sn and T*n are associated for each n, then we show that the sequence. S1·Sn(Tn·T1f) p r. converges a.e. for each nonnegative Lp-function f. This result includes Rota's "Alternierende Verfahren" theorem and its subsequent generalizations and covers new cases.

Original languageEnglish (US)
Pages (from-to)249-290
Number of pages42
JournalAdvances in Mathematics
Volume87
Issue number2
DOIs
StatePublished - Jun 1991

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