Counterexamples for interpolation of compact Lipschitz operators

TL;DR

This paper constructs examples demonstrating that the interpolation of compact Lipschitz operators between Banach couples does not always preserve compactness, contrasting with the linear case.

Contribution

It provides counterexamples showing that the compactness of Lipschitz operators is not necessarily maintained under interpolation.

Findings

Counterexamples show non-compactness in general

Interpolation does not always preserve compactness for Lipschitz operators

Contrast with linear case where compactness is preserved

Abstract

Let (A_0,A_1) and (B_0,B_1) be Banach couples with A_0 contained in A_1 and B_0 contained in B_1. Let T:A_1 --> B_1 be a possibly nonlinear operator which is a compact Lipschitz map of A_j into B_j for j=0,1. It is known that T maps the Lions-Peetre space (A_0,A_1)_\theta,q boundedly into (B_0,B_1)_\theta,q for each \theta in (0,1) and each q in [1,\infty), and that this map is also compact if if T is linear. We present examples which show that in general the map T:(A_0,A_1)_\theta,q --> (B_0,B_1)_\theta,q is not compact.