Journal of Inequalities and Applications
Volume 5 (2000), Issue 3, Pages 227-261

Interpolation of compact non-linear operators

A. J. G. Bento1,2

1Departamento de Matemática/Informática, Universidade da Beira Interior, Covilhã 6200, Portugal
2p/g pigeonholes, School of Mathematical Sciences, University of Sussex, Falmer, East Sussex, Brighton BN1 9QH, UK

Received 19 May 1999; Revised 7 July 1999

Copyright © 2000 A. J. G. Bento. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let (E0,E1) and (F0,F1) be two Banach couples and let T:E0+E1F0+F1 be a continuous map such that T:E0F0 is a Lipschitz compact operator and T:E1F1 is a Lipschitz operator. We prove that if T:E1F1 is also compact or E1 is continuously embedded in E0 or F1 is continuously embedded in F0, then T:(E0,E1)θ,q(F0,F1)θ,q is also a compact operator when 1q< and 1<θ<1. We also investigate the behaviour of the measure of non-compactness under real interpolation and obtain best possible compactness results of Lions–Peetre type for non-linear operators. A two-sided compactness result for linear operators is also obtained for an arbitrary interpolation method when an approximation hypothesis on the Banach couple (F0,F1) is imposed.