Noncompactness and noncompleteness in isometries of Lipschitz spaces |
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Authors: | Jesús Araujo Luis Dubarbie |
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Institution: | Departamento de Matemáticas, Estadística y Computación, Facultad de Ciencias, Universidad de Cantabria, Avenida de los Castros s/n, E-39071, Santander, Spain |
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Abstract: | We solve the following three questions concerning surjective linear isometries between spaces of Lipschitz functions Lip(X,E) and Lip(Y,F), for strictly convex normed spaces E and F and metric spaces X and Y:- (i)
- Characterize those base spaces X and Y for which all isometries are weighted composition maps.
- (ii)
- Give a condition independent of base spaces under which all isometries are weighted composition maps.
- (iii)
- Provide the general form of an isometry, both when it is a weighted composition map and when it is not.
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. |
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Keywords: | Linear isometry Banach-Stone Theorem Vector-valued Lipschitz function Biseparating map |
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