We study hermitian operators and isometries on spaces of vector-valued Lipschitz maps with the sum norm. 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.

中文翻译：

---

向量值 Lipschitz 映射空间上的等距和厄米算子

我们研究了具有总和范数的向量值 Lipschitz 映射空间上的厄米算子和等距。本文有两个主要定理。首先，我们证明每个厄米算子 $\operatorname {Lip}(X,E)$ ， 在哪里乙是复Banach空间，是广义复合算子。其次，我们给出了单位满射复线性等距的完整描述 $\operatorname {Lip}(X,\mathcal {A})$ ， 在哪里 $\mathcal {A}$ 是一个单位因子 $C^{*}$ -代数。这些结果改进了作者之前所说的结果。