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

TL;DR

This paper investigates the properties of certain sequence space couples, establishing conditions under which they exhibit specific interpolation characteristics and solving related open problems in the theory.

Contribution

It characterizes when the couples $(\,\ell^{p},\ell^{q})$ are Calderón-Mityagin and describes their interpolation orbits, providing a solution to a known open problem.

Findings

$(\ell^{p},\ell^{q})$ is Calderón-Mityagin iff $q\ge1$

Identifies interpolation orbits for all $p,q$ with $0\le p<q\le\infty$

Provides a positive solution to the Levitina-Sukochev-Zanin problem

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.