Interpolation of compact Lipschitz operators

Alon Ivtsan, Michael Cwikel, Eitan Tadmor

Publication date: 21 July 2009

Abstract: Let (A_0,A_1) and (B_0,B_1) be Banach couples such that A_0 is contained in A_1 and (B_0,B_1) satisfies Arne Persson's approximation condition (H). Let T:A_1 --> B_1 be a possibly nonlinear Lipschitz mapping which also maps A_0 into B_0 and satisfies the following quantitative compactnesss condition: Ta in ||a||_{A_0} K for each a in A_0, where K is a fixed compact subset of B_0. We show that T maps the real interpolation space (A_0,A_1)_{ heta,p} compactly into its counterpart (B_0,B_1)_{ heta,p} for each heta in (0,1) and p in [1,infty].

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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