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

Abstract

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 Lpfalse(μfalse), we show (M. Pierre's result) that A(λ−A)−1 is a “contraction on the positive cone”, i.e. Afalse(λ−Afalse)−1x≤x for all x∈L+pfalse(μfalse)false(λ>0false), 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 L1false(μfalse) spaces or in preduals of hermitian part of von Neumann algebras), we show that Nfalse(u−Cufalse)≤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.

An inequality in noncommutative L-spaces

“…We provide an alternative proof when p ∈ [3,4]. Denote by R x and L x the right and left multiplication operators by x ∈ M defined on all L p (M) (1 p ∞).…”

“…Assuming M finite and a = b + δ, b ∈ M ++ as above, the alternative proof when p ∈ [3,4] relies on Lemma 2.2:…”

“…Remark 2.6. The inequality (2) does not hold for p < 2; a counterexample with M = ℓ 2 ∞ can be found in [3]. There, a slight extension of ( 2) is given; one can replace E by any positive contractive projection C on L p (M).…”

“…In [3], Mustapha Mokhtar-Kharroubi notices that this inequality may be used to get contractivity results on the positive cone of L p (Ω, µ). This note originates from the question whether its noncommutative analogue remains true.…”

“…We hope that the techniques involved there may be useful for further studies. We end up by making explicit some results from [3] for noncommutative L p -spaces. We refer the reader to [4] for the definitions of L p -spaces associated to semifinite von Neumann algebras or more general ones.…”