This is an announcement for the paper "Noncompactness and noncompleteness in isometries of Lipschitz spaces" by Jesus Araujo and Luis Dubarbie.
Abstract: We solve the following two questions concerning surjective linear isometries between spaces of Lipschitz functions $\mathrm{Lip}(X,E)$ and $\mathrm{Lip}(Y,F)$, for strictly convex normed spaces $E$ and $F$ and metric spaces $X$ and $Y$: \begin{enumerate} \item Characterize those base spaces $X$ and $Y$ for which all isometries are weighted composition maps. \item Give a condition independent of base spaces under which all isometries are weighted composition maps. \end{enumerate} In particular, we prove that requirements of completeness on $X$ and $Y$ are not necessary when $E$ and $F$ are not complete, which is in sharp contrast with results known in the scalar context. We also give the special form of this kind of isometries.
Archive classification: math.FA
Mathematics Subject Classification: 2010: 47B33 (Primary), 46B04, 46E15, 46E40, 47B38 (Secondary)
Remarks: 14 pages, no figures, \documentclass[12pt]{amsart}
Submitted from: araujoj@unican.es
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1006.2995
or