Abstract
In this paper we investigate the connection between the range of nearest point projections in Lp -spaces and monotony properties of the projection operator. We show e.g. that a nearest point projection onto a closed convex subset of an Lp -space (1<p<∞) is monotone if and only if the closed convex range is a lattice. If the range is closed linear instead of closed convex then it turns out that positivity of the projection operator implies monotony, although the projection is in general not a linear operator. We can apply these results to a lot of known cases and to a case, in which the monotony of the projection operator was unknown up to now.
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ANDO,T. and AMEMIYA,I.: Almost everywhere convergence of prediction sequences in Lp (1<p<∞), Z. Wahrscheinlichkeitstheorie verw.Geb.4, 113–120 (1965)
BARLOW, BARTHOLOMEW, BREMNER, BRUNK: Statistical inference under order restrictions, Wiley, London-New York 1970
BRUNK,H.D.: Uniform inequalities for conditional p-means given a σ-lattice, Ann. Prob.3, 1025–1030 (1975)
HEWITT,E. and Stromberg,K.: Real and abstract analysis, Berlin-Heidelberg-New York, Springer 1969
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Landers, D., Rogge, L. On projections and monotony in Lp -spaces. Manuscripta Math 26, 363–369 (1979). https://doi.org/10.1007/BF01170260
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DOI: https://doi.org/10.1007/BF01170260