# Interpolation of compact bilinear operators

## TL;DR

This paper studies how the compactness of bilinear operators behaves under interpolation of Banach spaces, providing a general framework and new theorems applicable to various interpolation methods.

## Contribution

It introduces a unified approach for analyzing the stability of compactness in bilinear operators across different interpolation schemes.

## Key findings

- Established a one-sided bilinear interpolation theorem for compactness.
- Applied the framework to Peetre's and real interpolation methods.
- Demonstrated the stability of compactness under these interpolation techniques.

## Abstract

We investigate the stability of compactness of bilinear operators acting on the product of interpolation of Banach spaces. We develop a general framework for such results and our method applies to abstract methods of interpolation in the senseof Aronszajn and Gagliardo. A key step is to show an one-sided bilinear interpolation theorem on compactness for bilinear operators on couples satisfying an approximation property. We show applications to general cases, including Peetre's method and the general real interpolation methods.