Convergence of subdiagonal Pad\'{e} approximations of $C_0$-semigroups

Let $(r_n{)}_{n\in \mathbb{N}}$ be the sequence of subdiagonal Pad'{e} approximations of the exponential function. We prove that for $-A$ the generator of a uniformly bounded $C_0$-semigroup $T$ on a Banach space $X$, the sequence $(r_n(-tA){)}_{n\in \mathbb{N}}$ converges strongly to $T(t)$ on $\text{D}(A^{\alpha })$ for $\alpha >\frac{1}{2}$. Local uniform convergence in $t$ and explicit convergence rates in $n$ are established. For specific classes of semigroups, such as bounded analytic or exponentially $\gamma$-stable ones, stronger estimates are proved. Finally, applications to the inversion of the vector-valued Laplace transform are given.