Arazy-Cwikel property for quasi-Banach couples

TL;DR

This paper extends the Arazy-Cwikel property to quasi-Banach spaces, showing it holds under certain conditions for classes of uniformly K-monotone spaces and interpolation quasi-Banach spaces, with implications for classical spaces.

Contribution

It establishes the Arazy-Cwikel property for quasi-Banach K-spaces and interpolation spaces under new conditions, broadening its applicability in quasi-Banach space theory.

Findings

The property holds for uniformly K-monotone spaces with mutually closed couples.

It applies to all quasi-Banach K-spaces with respect to mutually closed Banach couples.

It also holds for interpolation quasi-Banach spaces when couples have the uniform Calderón-Mityagin property.

Abstract

The main result of this paper establishes that the known Arazy-Cwikel property holds for classes of uniformly K-monotone spaces in the quasi-Banach setting provided that the initial couple is mutually closed. As a consequence, we get that the class of all quasi-Banach K-spaces (i.e., interpolation spaces which are described by the real K-method) with respect to an arbitrary mutually closed Banach couple enjoys the Arazy-Cwikel property. Another consequence complements some previous results by Bykov and Ovchinnikov, showing that this property holds also for the class of all interpolation quasi-Banach spaces with respect to a quasi-Banach couple whenever all the couples involved have the uniform Calder\'{o}n-Mityagin property. We apply these results to some classical families of spaces.