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

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 spacesL p ( μ ) , we show (M. Pierre's result) that A ( λ − A ) − 1is a “ contraction on the positive cone ”, i.e.A ( λ − A ) − 1 x ≤ xfor 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 realL 1 ( μ )spaces or in preduals of hermitian part of von Neumann algebras), we show that N ( u − C u ) ≤ ufor 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.