Interpolation of compact nonlinear operators
Let \((E_{0},E_{1})\) and \((F_{0},F_{1})\) be two Banach couples and let \(T:E_{0}+E_{1}\to F_{0}+F_{1}\) be a continuous map such that \(T:E_{0}\to F_{0}\) is a Lipschitz compact operator and \(T_{1}:E_{1}\to F_{1}\) is a Lipschitz operator. In this paper it is proved that if \(T:E_{1}\to F_{1}\) is also compact or \(E_{1}\) is continuously embedded in \(E_{0}\) or \(F_{1}\) is continuously embedded in \(F_{0}\), then \(T:(E_{0},E_{1})_{\theta, q}\to (F_{0},F_{1})_{\theta, q}\) is also a compact operator when \(1\leq q<\infty\) and \(0<\theta <1.\) Also the measure of non-compactness under real interpolation is investigated. The results of this paper extend some theorems of Cobos et al.
scientific article; zbMATH DE number 1507754