General Theorem on Interpolation of Compact Operators

TL;DR

This paper presents a general theorem ensuring the preservation of compactness of linear operators during interpolation in Banach spaces, applicable to various spaces and interpolation methods without requiring detailed analytical operator descriptions.

Contribution

It introduces an abstract, broadly applicable theorem on compactness preservation under interpolation, relying on basic properties of compact sets and bases, avoiding complex analytical assumptions.

Findings

Proves compactness preservation for a wide class of interpolation functors.

Applicable to arbitrary Banach spaces and interpolation methods, including complex interpolation.

Includes a corollary on two-sided compactness.

Abstract

We prove an abstract theorem on keeping the compactness property of a linear operator after interpolation in Banach spaces. No analytical presentation of operators, spaces and interpolation functor is required. We use only some little-known properties of compact sets and various facts about bases and basic sequences (with detailed references to the monograph "Bases in Banach spaces", Vol. I-II by I.M. Singer). Therefore the results are applicable to arbitrary spaces and any interpolation functor, including the complex method. The "two-sided" compactness is also mentioned at the end of this paper as a mere corollary.