Arazy-Cwikel and Calderón-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 Calderón-Mityagin couple if and only if $q\ge 1$. 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 Calderón-Mityagin property or not.

References in corpus (2)

• Cwikel estimates revisited
• A description of interpolation spaces for quasi-Banach couples by real $K$-method

Cited by in corpus (2)

• A description of interpolation spaces for quasi-Banach couples by real $K$-method
• Lorentz-Shimogaki-Arazy-Cwikel Theorem Revisited