Using a recently developed $\mathcal H$-calculus we propose a unified approach to the study of rational approximations of holomorphic semigroups on Banach spaces. We provide unified and simple proofs to a number of basic results on semigroup approximations and substantially improve some of them. We show that many of our estimates are essentially optimal, thus complementing the existing literature.

翻译：利用近期发展的$\mathcal H$-演算，我们提出了一种统一的方法来研究Banach空间上全纯半群的有理逼近。我们对半群逼近的一系列基本结果给出了统一且简洁的证明，并显著改进了其中部分结论。我们证明了我们的许多估计本质上是最优的，从而对现有文献进行了补充。