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

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].