A Survey on Isometries Between Lipschitz Spaces

Abstract

The famous Banach-Stone theorem, which characterizes surjective linear isometries between C(X) spaces as certain weighted composition operators, has motivated the study of isometries defined on different function spaces. The research on surjective linear isometries between spaces of Lipschitz functions is a subject of long tradition which goes back to the sixties with the works of de Leeuw and Roy, and followed by those by Mayer-Wolf, Weaver, Araujo and Dubarbie, and Botelho, Fleming and Jamison. This topic continues to attract the attention of some authors. In the setting of Lipschitz spaces, we present a survey on non-necessarily surjective linear isometries and codimension 1 linear isometries, vector-valued linear isometries, local isometries and generalized bi-circular projections, 2-local isometries, projections and averages of isometries and hermitian operators. We also raise some open problems on bilinear isometries and approximate isometries in the same context.