Convergence of subdiagonal Padé approximations of $C_{0}$-semigroups

Let $(r_{n})_{n \in \mathbb{N}}$ be the sequence of subdiagonal Padé 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 $\textrm{D}(A^α)$ for $α>\frac{1}{2}$. Local uniform convergence in $t$ and explicit convergence rates in $n$ are established. For specific classes of semigroups, such as bounded analytic or exponentially $γ$-stable ones, stronger estimates are proved. Finally, applications to the inversion of the vector-valued Laplace transform are given.

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