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Pointwise Convergence of Alternating Sequences

Published online by Cambridge University Press:  20 November 2018

M. A. Akcoglu  and
L. Sucheston
M. A. Akcoglu
Affiliation:
University of Toronto, Toronto, Ontario
L. Sucheston
Affiliation:
The Ohio State University, Columbus, Ohio
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Let 1 < p < ∞ and let Lp be the usual Banach Space of complex valued functions on a σ-finite measure space. Let (Tn), n ≧ 1, be a sequence of positive linear contractions on Lp. Hence and , where is the part of Lp that consists of non-negative Lp functions. The adjoint of Tn is denoted by which is a positive linear contraction of Lq with q = p/(p — 1).

Our purpose in this paper is to show that the alternating sequences associated with (Tn), as introduced in [2], converge almost everywhere. Complete definitions will be given later. When applied to a non negative function, however, this result is reduced to the following theorem.

(1.1) THEOREM. If (Tn) is a sequence of positive contractions of Lp then (1.2)

exists a.e. for all.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 40 , Issue 3 , 01 June 1988 , pp. 610 - 632
Copyright
Copyright © Canadian Mathematical Society 1988

References

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