We improve the classical results by Brenner and Thom\'ee 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 Pad\'e approximations of operator semigroups. Our approach is direct and based on the theory of the $\mathcal B$- functional calculus developed recently. On the way, we elaborate a new and simple approach to construction of the $\mathcal B$-calculus thus making the paper essentially self-contai

翻译：我们改进了Brenner和Thomée关于算子半群有理逼近的经典结果。在希尔伯特空间框架下，我们引入了初始数据更精细的正则性标度，给出了更优的稳定性估计，并获得了最优逼近速率。此外，我们还强化了Egert-Rozendaal关于算子半群次对角Padé逼近的结果。我们的方法直接且基于近期发展的$\mathcal B$-函数演算理论。在此过程中，我们阐述了一种构建$\mathcal B$-演算的简洁新方法，从而使本文基本上成为自封闭的。