Arazy-Cwikel and Calder\'on-Mityagin type properties of the couples $(\ell^{p},\ell^{q})$, $0 \le p<q\le\infty$

Abstract

We establish Arazy-Cwikel type properties for the family of couples $(\ell^{p},\ell^{q})$, $0\le p<q\le\infty$, and show that $(\ell^{p},\ell^{q}) $ is a Calder\'on-Mityagin couple if and only if $q\ge1$. Moreover, we identify interpolation orbits of elements with respect to this couple for all $p$ and $q$ such that $0\le p<q\le\infty$ and obtain a simple positive solution of a Levitina-Sukochev-Zanin problem, clarifying its connections with whether $(\ell^{p},\ell^{q})$ has the Calder\'on-Mityagin property or not.