2-local isometries on vector-valued Lipschitz function spaces

Ya-Shu Wang

Department of Applied Mathematics

National Chung Hsing University


    Let `E` and `F` be Banach spaces. Let `\mathcal{S}` be a subset of the space `L(E,F)` of all continuous linear maps from `E` into `F`. A (non-necessarily linear nor continuous) mapping `\Delta:E\to F` is a 2-local `\mathcal{S}`-map, if for any `x,y\in E`, there exists `T_{x,y}\in\mathcal{S}`, depending on `x` and `y`, such that
`\Delta(x)=T_{x,y}(x)" and "\Delta(y)=T_{x,y}(y).`

In this talk, I will present a description of the 2-local isometries on the Banach space `"Lip"(X,E)` of vector-valued Lipschitz functions from compact metric space `X` to Banach space `E` in terms of a generalized composition operator. Also, I will show when every 2-local (standard) isometry on `"Lip"(X,E)` is both linear and surjective.
`bb"Co-author(s):"`Jiménez-Vargas, L. Li, A. M. Peralta and L. Wang.