Journal of Inequalities and Applications
Volume 5 (2000), Issue 3, Pages 227-261
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.
Let and be two Banach couples and let be a continuous map such that is a Lipschitz compact operator and is a Lipschitz operator. We prove that if is also compact or is continuously embedded in or is continuously embedded in , then is also a compact operator when and . 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 is imposed.