Journal of Inequalities and Applications 
Volume 5 (2000), Issue 3, Pages 227-261
doi:10.1155/S1025583400000126

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

Abstract

Let (E0,E1) and (F0,F1) be two Banach couples and let T:E0+E1→F0+F1 be a continuous map such that T:E0→F0 is a Lipschitz compact operator and T:E1→F1 is a Lipschitz operator. We prove that if T:E1→F1 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 1≤q<∞ 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.