The Koopman linearization of measure-preserving systems or topological dynamical systems on compact spaces has proven to be extremely useful. In this article, we look at dynamics given by continuous semiflows on completely regular spaces, which arise naturally from solutions of PDEs. We introduce Koopman semigroups for these semiflows on spaces of bounded continuous functions. As a first step we study their continuity properties as well as their infinitesimal generators. We then characterize them algebraically (via derivations) and lattice theoretically (via Kato’s equality). Finally, we demonstrate—using the example of attractors—how this Koopman approach can be used to examine properties of dynamical systems. This article is part of the theme issue ‘Semigroup applications everywhere’.

中文翻译：

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完全规则空间上动力系统的 Koopman 理论

紧凑空间上的测量保持系统或拓扑动力系统的 Koopman 线性化已被证明是非常有用的。在本文中，我们将研究完全规则空间上的连续半流给出的动力学，这些动力学是从偏微分方程的解中自然产生的。我们在有界连续函数的空间上为这些半流引入 Koopman 半群。作为第一步，我们研究它们的连续性特性以及它们的无穷小发生器。然后我们用代数（通过推导）和格子理论（通过加藤等式）来表征它们。最后，我们使用吸引子的例子演示了如何使用这种 Koopman 方法来检查动态系统的属性。这篇文章是主题问题“无处不在的半组应用程序”的一部分。