Counterexamples for interpolation of compact Lipschitz operators

Details Counterexamples for interpolation of compact Lipschitz operators Counterexamples for interpolation of compact Lipschitz operators

2009-06-12

arxiv.org/abs/0906.2432v1

Mathematics

Functional Analysis

22 pages. The main results are on pages 1-8. Later pages contain some additional more elaborate counterexamples

Scientific paper

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.

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