A unified approach to compatibility theorems on invertible interpolated operators

TL;DR

This paper establishes the stability of isomorphisms between Banach spaces via various interpolation methods, leading to new results on operator equations, inverse uniqueness, and spectral properties in the context of interpolated operators.

Contribution

It provides a unified proof of compatibility theorems for multiple interpolation methods, extending stability, invertibility, and spectral analysis results for interpolated operators.

Findings

Proves stability of isomorphisms across several interpolation methods.

Shows existence of solutions to certain operator equations.

Establishes connections between spectra of interpolated operators.

Abstract

We prove the stability of isomorphisms between Banach spaces generated by interpolation methods introduced by Cwikel-Kalton-Milman-Rochberg which includes, as special cases, the real and complex methods up to equivalence of norms and also the so-called $\pm$ or $G_1$ and $G_2$ methods defined by Peetre and Gustavsson-Peetre. This result is used to show the existence of solution of certain operator analytic equation. A by product of these results is a more general variant of the Albrecht-M\"uller result which states that the interpolated isomorphisms satisfy uniqueness-of-inverses between interpolation spaces. We show applications for positive operators between Calder\'on function lattices. We also derive connections between the spectrum of interpolated operators.