An intriguing feature of positive \(C_0\)-semigroups on function spaces (or more generally on Banach lattices) is that their long-time behaviour is much easier to describe than it is for general semigroups. In particular, the convergence of semigroup operators (strongly or in the operator norm) as time tends to infinity can be characterized by a set of simple spectral and compactness conditions. In the present paper, we show that similar theorems remain true for the larger class of (uniformly) eventually positive semigroups—which recently arose in the study of various concrete differential equations. A major step in one of our characterizations is to show a version of the famous Niiro–Sawashima theorem for eventually positive operators. Several proofs for positive operators and semigroups do not work in our setting any longer, necessitating different arguments and giving our approach a distinct flavour.

正\(C_0\) 的一个有趣特征函数空间上的 -半群（或更一般地在 Banach 格上）是它们的长期行为比一般半群更容易描述。特别是，随着时间趋于无穷大，半群算子（强或在算子范数中）的收敛可以用一组简单的谱和紧性条件来表征。在本文中，我们展示了类似的定理对于更大的（一致的）最终正半群仍然成立——它们最近出现在各种具体微分方程的研究中。我们描述的一个重要步骤是为最终正算子展示著名的 Niiro-Sawashima 定理的一个版本。一些正算子和半群的证明不再适用于我们的设置，