The notion Perron–Frobenius theory usually refers to the interaction between three properties of operator semigroups: positivity, spectrum and long-time behaviour. These interactions gives rise to a profound theory with plenty of applications. By a brief walk-through of the field and with many examples, we highlight two aspects of the subject, both related to the long-time behaviour of semigroups: (i) The classical question how positivity of a semigroup can be used to prove convergence to an equilibrium as t → ∞. (ii) The more recent phenomenon that positivity itself sometimes occurs only for large t, while being absent for smaller times. This article is part of the theme issue ‘Semigroup applications everywhere’.

中文翻译：

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正不可约半群及其长期行为

Perron-Frobenius 理论的概念通常是指算子半群的三个属性之间的相互作用：正性、谱和长期行为。这些相互作用产生了具有大量应用的深刻理论。通过对该领域的简要介绍和许多例子，我们强调了该主题的两个方面，都与半群的长期行为有关：（i）经典问题如何使用半群的正性来证明收敛达到 t → ∞ 的平衡。(ii) 最近的现象，即积极性本身有时只在大 t 时出现，而在较小的时间里不存在。这篇文章是主题问题“无处不在的半组应用程序”的一部分。