Isometries and hermitian operators on spaces of vector-valued Lipschitz maps

Abstract

We study hermitian operators and isometries on spaces of vector-valued Lipschitz maps with the sum norm: $\Vert \cdot {\Vert }_{\infty }+L(\cdot )$. There are two main theorems in this paper. Firstly, we prove that every hermitian operator on $\mathrm{Lip}(X,E)$, where $E$ is a complex Banach space, is a generalized composition operator. Secondly, we give a complete description of unital surjective complex linear isometries on $\mathrm{Lip}(X,\mathcal{A})$ where $\mathcal{A}$ is a unital factor $C^{\ast }$-algebra. These results improve previous results stated by the author.