Contractivity theorems in real ordered Banach spaces with applications to relative operator bounds, ergodic projections and conditional expectations
Résumé
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 (), we show (M. Pierre's result) that A(A) 1 is a "contraction on the positive cone", i.e. A(A) 1 x kxk for all x 2 L p + () (> 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 L 1 () spaces or in preduals of hermitian part of von Neumann algebras), we show that N (u Cu) kuk 8u 2 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.
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