A new approach to interpolation of compact linear operators

Abstract

We prove an abstract theorem on keeping the compactness property of a linear operator after interpolation in Banach spaces. Our approach consists of two features. Applying the principle "reductio ad absurdum," we obtain a possibility to carry out all proofs only for some specially constructed subspaces of the given spaces, e.g., having a common Schauder basis. As a second feature, we consider in all assertions only embedding operators obtaining the full result just at the end of the paper. No analytical presentation of operators, spaces and interpolation functors is required and the complex method is admissible as a particular case.