Counterexamples for interpolation of compact Lipschitz operators
Abstract
Let (A0,A1) and (B0,B1) be Banach couples with A0 contained in A1 and B0 contained in B1. Let T:A1 --> B1 be a possibly nonlinear operator which is a compact Lipschitz map of Aj into Bj for j=0,1. It is known that T maps the Lions-Peetre space (A0,A1)θ,q boundedly into (B0,B1)θ,q for each θ in (0,1) and each q in [1,∞), and that this map is also compact if if T is linear. We present examples which show that in general the map T:(A0,A1)θ,q --> (B0,B1)θ,q is not compact.
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