An alternative characterization of normed interpolation spaces between ${\ell }^1$ and ${\ell }^q$

Abstract

Given a constant $q\in (1,\infty )$, we study the following property of a normed sequence space $E$: ===================== If ${\left\{x_n\right\}}_{n\in \mathbb{N}}$ is an element of $E$ and if ${\left\{y_n\right\}}_{n\in \mathbb{N}}$ is an element of ${\ell }^q$ such that ${\sum }_{n=1}^{\infty }{\left|x_n\right|}^q={\sum }_{n=1}^{\infty }{\left|y_n\right|}^q$ and if the nonincreasing rearrangements of these two sequences satisfy ${\sum }_{n=1}^N{\left|x_n^{\ast }\right|}^q\le {\sum }_{n=1}^N{\left|y_n^{\ast }\right|}^q$ for all $N\in \mathbb{N}$, then ${\left\{y_n\right\}}_{n\in \mathbb{N}}\in E$ and ${\Vert {\left\{y_n\right\}}_{n\in \mathbb{N}}\Vert }_E\le C{\Vert {\left\{x_n\right\}}_{n\in \mathbb{N}}\Vert }_E$ for some constant $C$ which depends only on $E$. ===================== We show that this property is very close to characterizing the normed interpolation spaces between ${\ell }^1$ and ${\ell }^q$. More specificially, we first show that every space which is a normed interpolation space with respect to the couple $\left({\ell }^p,{\ell }^q\right)$ for some $p\in [1,q]$ has the above mentioned property. Then we show, conversely, that if $E$ has the above mentioned property, and also has the Fatou property, and is contained in ${\ell }^q$, then it is a normed interpolation space with respect to the couple $\left({\ell }^1,{\ell }^q\right)$. These results are our response to a conjecture of Galina Levitina, Fedor Sukochev and Dmitriy Zanin in arXiv:1703.04254v1 [math.OA].

33 pages