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

TL;DR

This paper characterizes normed interpolation spaces between  and  using a property related to rearrangements and shows its equivalence to being an interpolation space under certain conditions, addressing a conjecture in the field.

Contribution

It introduces a new property that characterizes normed interpolation spaces between  and  and proves its equivalence to the classical interpolation space concept under additional assumptions.

Findings

The property is necessary for spaces that are interpolation spaces between  and .

Spaces with the property and the Fatou property are interpolation spaces between  and .

The results confirm a conjecture by Levitina, Sukochev, and Zanin.

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 $\sum_{n=1}^{N}\left|x_{n}^{*}\right|^{q}\le\sum_{n=1}^{N}\left|y_{n}^{*}\right|^{q}$ for all $N\in\mathbb{N}$, then $\left\{ y_{n}\right\}_{n\in\mathbb{N}}\in E$ and $\left\Vert \left\{ y_{n}\right\}_{n\in\mathbb{N}}\right\Vert_{E}\le C\left\Vert \left\{ x_{n}\right\}_{n\in\mathbb{N}}\right\Vert_{E}$ 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].