TY - JOUR

T1 - Representation of positive operators and alternating sequences

AU - Akcoglu, M. A.

AU - Baxter, J. R.

AU - Lee, W. M.F.

N1 - Copyright:
Copyright 2018 Elsevier B.V., All rights reserved.

PY - 1991/6

Y1 - 1991/6

N2 - 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.

AB - 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.

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U2 - 10.1016/0001-8708(91)90073-G

DO - 10.1016/0001-8708(91)90073-G

M3 - Article

AN - SCOPUS:0039650850

VL - 87

SP - 249

EP - 290

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 2

ER -