论文标题
关于运算符值功能的最终规律性属性
On Eventual Regularity Properties of Operator-Valued Functions
论文作者
论文摘要
令$ \ MATHCAL {l}（x; y）$为有限线性运算符的空间，从Banach Space $ x $到Banach Space $ y $。给定一个运算符值$ u：\ mathbb {r} _ {\ geq 0} \ rightarrow \ mathcal \ mathcal {l}（x; y）$，假设每个orbit $ t \ mapsto u（t）x $具有常规性属性（例如，contranition，directibal of in Intever $ in Intever $ in Intever $ in Intevely $ in Interve $ in Interve $（ x $。在本文中，我们基于Baire-type参数开发一个抽象的设置，该设置允许在某些条件下系统地删除对$ x $的依赖。之后，我们将此理论框架应用于半群理论中同样感兴趣的几种不同的规律性属性。特别是，对强烈连续函数的最终可区分性的概括$ u：\ mathbb {r} _ {\ geq 0} \ rightarrow \ rightArrow \ mathcal {l}（x; y）$由Iley和Bárta获得，这是我们方法的特殊情况。
Let $\mathcal{L}(X;Y)$ be the space of bounded linear operators from a Banach space $X$ to a Banach space $Y$. Given an operator-valued function $u:\mathbb{R}_{\geq 0}\rightarrow \mathcal{L}(X;Y)$, suppose that every orbit $t\mapsto u(t)x$ has a regularity property (e.g. continuity, differentiability, etc.) on some interval $(t_x,\infty)$ in general depending on $x\in X$. In this paper we develop an abstract set-up based on Baire-type arguments which allows, under certain conditions, removing the dependency on $x$ systematically. Afterwards, we apply this theoretical framework to several different regularity properties that are of interest also in semigroup theory. In particular, a generalisation of the prior results on eventual differentiability of strongly continuous functions $u:\mathbb{R}_{\geq 0} \rightarrow \mathcal{L}(X;Y)$ obtained by Iley and Bárta follows as a special case of our method.
