Counterexamples for interpolation of compact Lipschitz operators

Publication date: 12 June 2009

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)_ heta,q boundedly into (B_0,B_1)_ heta,q for each heta 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)_ heta,q --> (B_0,B_1)_ heta,q is not compact.

Mathematics Subject Classification ID

Interpolation between normed linear spaces (46B70) Compactness in Banach (or normed) spaces (46B50) Nonlinear operators and their properties (47H99)

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