# Two remarks on the interpolation space

## TL;DR

This paper investigates properties of interpolation spaces, establishing equalities between certain spaces and constructing isomorphisms that relate different interpolation couples, advancing understanding of their structure.

## Contribution

It demonstrates the equality of two interpolation spaces involving measures and sequence spaces, and constructs specific isomorphisms between interpolation spaces introduced by Garling-Smith.

## Key findings

- Proves $(M(\
- ") ext{ and }c_0( ext{Z}))_ heta = (L^1,c_0( ext{Z}))_ heta$ for $0< heta<1$.
- Constructs isomorphisms $U_ heta$ between interpolation spaces that preserve structure and relate different couples.

## Abstract

Dans ce travail, on montre que $(M(\mathbb{T}),c_0(\mathbb{Z}))_\theta = (L^1,c_0(\mathbb{Z}))_\theta$, $0<\theta <1$. Dans la suite on montre pour le couple d'interpolation $(C_0,C_1)$ trouv\'e par Garling-Smith qu'il existe un isomorphisme $U_\theta: (C_0,C_0+C_1)_{\theta ,p}\rightarrow (C_1,C_0+C_1)_{\theta, p}$ (resp. $U_\theta : (C_0,C_0+C_1)_\theta \rightarrow (C_1,C_0+C_1)_\theta)$ tel que sa restriction \`a $C_{\theta, p}$ (resp. \`a $C_\theta)$ est un isomorphisme : $C_{\theta, p} \rightarrow C_{1-\theta, p}$ (resp. $C_\theta \rightarrow C_{1-\theta })$. -- In this work we show that $(M(\mathbb{T}),c_0(\mathbb{Z}))_\theta = (L^1,c_0(\mathbb{Z}))_\theta$, $0<\theta <1.$ In the following we show for the interpolation couple found by Garling-Smith that there exists an isomorphism $U_\theta: (C_0,C_0+C_1)_{\theta ,p}\rightarrow (C_1,C_0+C_1)_{\theta, p}$ (resp. $U_\theta : (C_0,C_0+C_1)_\theta \rightarrow (C_1,C_0+C_1)_\theta)$ such that its restriction to $C_{\theta ,p}$ (resp. to $C_\theta)$ is an isomorphism : $C_{\theta, p} \rightarrow C_{1-\theta, p}$ (resp. $C_\theta \rightarrow C_{1-\theta })$.