论文标题
非线性双向图
Nonlinear biseparating maps
论文作者
论文摘要
如果$ t $是两次射击，则据说在矢量值函数的空间之间作用的加性地图$ t $在且仅当$ f $ and $ g $之间是脱节的，并且仅当$ f $ and $ g $是脱节的，并且仅当$ tf $和$ tf $ and $ tg $是分离的。请注意，添加剂双眼保留$ \ Mathbb {q} $ - 线性。对于一般的非线性地图$ t $，上面给出的双分裂的定义事实证明太弱了，无法确定$ t $的结构。在本文中，我们提出了对矢量值函数空间之间作用的一般非线性操作员的分配图的修订定义，该函数与添加剂图的先前定义相吻合。在涉及功能空间的一些轻度假设下，事实证明，当且仅当局部确定它时，地图是分配的。然后，我们深入研究一些特定的函数空间 - 连续函数的空间，均匀连续的函数和Lipschitz函数 - 并表征作用于它们的双层图。作为副产品，获得了某些形式的自动连续性。在均匀连续和Lipschitz函数的情况下，我们还证明了分配图的一些更好的特性。
An additive map $T$ acting between spaces of vector-valued functions is said to be biseparating if $T$ is a bijection so that $f$ and $g$ are disjoint if and only if $Tf$ and $Tg$ are disjoint. Note that an additive bijection retains $\mathbb{Q}$-linearity. For a general nonlinear map $T$, the definition of biseparating given above turns out to be too weak to determine the structure of $T$. In this paper, we propose a revised definition of biseparating maps for general nonlinear operators acting between spaces of vector-valued functions, which coincides with the previous definition for additive maps. Under some mild assumptions on the function spaces involved, it turns out that a map is biseparating if and only if it is locally determined. We then delve deeply into some specific function spaces -- spaces of continuous functions, uniformly continuous functions and Lipschitz functions -- and characterize the biseparating maps acting on them. As a by-product, certain forms of automatic continuity are obtained. We also prove some finer properties of biseparating maps in the cases of uniformly continuous and Lipschitz functions.
