We improve the classical results by Brenner and Thomée on rational approximations of operator semigroups. In the setting of Hilbert spaces, we introduce a finer regularity scale for initial data, provide sharper stability estimates, and obtain optimal approximation rates. Moreover, we strengthen a result due to Egert-Rozendaal on subdiagonal Padé approximations of operator semigroups. Our approach is direct and based on the theory of the - functional calculus developed recently. On the way, we elaborate a new and simple approach to construction of the -calculus thus making the paper essentially self-contained.

中文翻译：

---

通过[公式省略]微积分算子半群的有理逼近

我们改进了 Brenner 和 Thomée 关于算子半群有理逼近的经典结果。在希尔伯特空间的设置中，我们为初始数据引入了更精细的正则性尺度，提供了更清晰的稳定性估计，并获得了最佳逼近率。此外，我们还强化了 Egert-Rozendaal 关于算子半群的下对角 Padé 近似的结果。我们的方法是直接的，并且基于最近发展的泛函微积分理论。在此过程中，我们详细阐述了一种新的、简单的方法来构造 β 微积分，从而使本文本质上是独立的。