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

TL;DR

This paper characterizes hermitian operators and isometries on spaces of vector-valued Lipschitz maps, showing they are generalized composition operators or specific algebraic isometries, advancing understanding of their structure.

Contribution

It provides a complete description of hermitian operators and surjective isometries on vector-valued Lipschitz spaces, extending previous results.

Findings

Hermitian operators are generalized composition operators.

Complete description of unital surjective isometries on Lipschitz spaces.

Improves upon previous characterizations in the literature.

Abstract

We study hermitian operators and isometries on spaces of vector-valued Lipschitz maps with the sum norm: $\|\cdot\|_{\infty}+L(\cdot)$. There are two main theorems in this paper. Firstly, we prove that every hermitian operator on $\operatorname{Lip}(X,E)$, 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 $\operatorname{Lip}(X,\mathcal{A})$ where $\mathcal{A}$ is a unital factor $C^{*}$-algebra. These results improve previous results stated by the author.