6533b822fe1ef96bd127d5b1

RESEARCH PRODUCT

Contractivity results in ordered spaces. Applications to relative operator bounds and projections with norm one

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This paper provides various “contractivity” results for linear operators of the form I−C where C are positive contractions on real ordered Banach spaces X. If A generates a positive contraction semigroup in Lebesgue spaces Lp(μ), we show (M. Pierre's result) that A(λ−A)−1 is a “contraction on the positive cone”, i.e. A(λ−A)−1x≤x for all x∈L+p(μ)(λ>0), provided that p⩾2.  We show also that this result is not true for 1 ⩽ p<2. We give an extension of M. Pierre's result to general ordered Banach spaces X under a suitable uniform monotony assumption on the duality map on the positive cone X+. We deduce from this result that, in such spaces, I−C is a contraction on X+ for any positive projection C with norm 1. We give also a direct proof (by E. Ricard) of this last result if additionally the norm is smooth on the positive cone. For any positive contraction C on base-norm spaces X (e.g. in real L1(μ) spaces or in preduals of hermitian part of von Neumann algebras), we show that N(u−Cu)≤u for all u∈X where N is the canonical half-norm in X. For any positive contraction C on order-unit spaces X (e.g. on the hermitian part of a C* algebra), we show that I−C is a contraction on X+. Applications to relative operator bounds, ergodic projections and conditional expectations are given.