2-local isometries on vector-valued Lipschitz function spaces

Ya-Shu Wang

Department of Applied Mathematics

National Chung Hsing University

yashu@nchu.edu.tw

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.