Rational approximation of operator semigroups via the $\mathcal{B}$-calculus

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 $\mathcal{B}$- 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