Journal of Inequalities and Applications
Volume 5 (2000), Issue 3, Pages 227-261
doi:10.1155/S1025583400000126
Interpolation of compact non-linear operators
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.
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.