Une remarque sur les espaces d'interpolation faiblement localement uniform\'ement convexes

TL;DR

This paper investigates the properties of interpolation spaces between Banach spaces, showing that under certain convexity conditions, the interpolation space coincides with the classical real interpolation space.

Contribution

It establishes a new link between weakly LUR properties of certain closures and the equality of interpolation spaces and their classical counterparts.

Findings

Under weakly LUR conditions, $A^ heta$ equals $A_ heta$ for all $ heta$.

Introduces a contraction mapping $R^ heta$ connecting different interpolation spaces.

Provides conditions under which interpolation spaces are strongly related to their duals.

Abstract

Let $(A_0, A_1)$ be an interpolation couple, and let $B_j$ be the closure of $A_0^\ast \cap A_1^\ast$ in $A_j^\ast$, $j = 0, 1$. For every $\theta \in \, ]0, 1[$, there exists a natural one to one contraction $R^\theta : A^\theta \rightarrow (B_0^\ast, B_1^\ast)^\theta$. For some $\beta \in \, ]0, 1[$, the closure of $R^\beta (A^\beta)$ in $(B_0^\ast, B_1^\ast)^\beta$ is supposed to be weakly LUR. Then $A^\theta = A_\theta$ for every $\theta \in \, ]0, 1[$.