We prove a Desch-Schappacher type perturbation theorem for one-parameter semigroups on Banach spaces which are not strongly continuous for the norm, but possess a weaker continuity property. In this paper we chose to work in the framework of bi-continuous semigroups. This choice has the advantage that we can treat in a unified manner two important classes of semigroups: implemented semigroups on the Banach algebra $\mathscr{L}(E)$ of bounded, linear operators on a Banach space $E$, and semigroups on the space of bounded and continuous functions over a Polish space induced by jointly continuous semiflows. For both of these classes we present an application of our abstract perturbation theorem

中文翻译：

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双连续半群的 Desch-Schappacher 摄动定理

我们证明了 Banach 空间上的单参数半群的 Desch-Schappacher 型摄动定理，这些半群对于范数不是强连续的，但具有较弱的连续性。在本文中，我们选择在双连续半群的框架中工作。这种选择的优点是我们可以统一处理两类重要的半群：在 Banach 代数 $\mathscr{L}(E)$ 上实现的半群，在 Banach 空间 $E$ 上的有界线性算子和半群关于由联合连续半流引起的波兰空间上有界和连续函数的空间。对于这两个类，我们提出了抽象扰动定理的应用