A discrete framework for the interpolation of Banach spaces

TL;DR

This paper introduces a unified discrete framework for interpolating Banach spaces, encompassing classical and recent methods, and extends key results across these methods using a sequential structure approach.

Contribution

It develops a novel discrete framework that generalizes real, complex, and newer interpolation methods for Banach spaces, enabling broader applicability and extension of existing results.

Findings

Unified framework for various interpolation methods

Extension of results to new interpolation techniques

Application to analytic operator families and intersections

Abstract

We develop a discrete framework for the interpolation of Banach spaces, which contains the well-known real and complex interpolation methods, but also more recent methods like the Rademacher, $\gamma$- and $\ell^q$-interpolation methods. Our framework is based on a sequential structure imposed on a Banach space, which allows us to deduce properties of interpolation methods from properties of sequential structures. Our framework has a formulation modelled after both the real and the complex interpolation methods. This enables us to extend various results, previously known only for either the real or the complex interpolation method, to all interpolation methods that fit into our framework. As applications, we prove an interpolation result for analytic operator families and an interpolation result for intersections.