Interpolation des opérateurs de Radon–Nikodym et des espaces L Λ p , h Λ p

Abstract

Let A 0 , A 1 be two Banach spaces, i:A 0 →A 1 a continuous injection with dense range and 0<α<β<1. In the first part of this work, we show that if i:A α,p →A β,p is a Radon–Nikodym operator, then A α,p has the Radon–Nikodym property. We show that this result is false for complex interpolation (Theorem 2.2). We also show that there are two Banach spaces B 0 ,B 1 and i:B 0 →B 1 a Radon–Nikodym injection such that for 0<α<β<1, i:A α →A β is not a Radon–Nikodym operator. We introduce the interpolation spaces A θ + , θ∈]0,1[, and show that A ¯ θ is an isometric subspace of A θ + .