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 spaces L-p(mu), we show (M. Pierre's result) that A(gamma-A)(-1) is a ``contraction on the positive cone'', i.e. parallel to A(lambda-A)(-1)x parallel to <= parallel to x parallel to for all x is an element of L-+(p)(mu)(lambda > 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 spacesX (e.g. in real L-1(mu) spaces or in preduals of hermitian part of von Neumann algebras), we show that N(u-Cu) <= parallel to u parallel to for all u is an element of 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. (C) 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim