Rational approximation schemes for bi-continuous semigroups (Q929576)

An operator semigroup \(T(t)_{t\geq 0}\) on a Banach space \(X\) is said to be of type \((M,\omega)\) if, for some \(M> 1\) and \(\omega\in\mathbb{R}\), one has that \(\| T(t)\|\leq Me^{\omega t}\) for all \(t\geq 0\). Such a semigroup is called bi-continuous with respect to some topology \(\tau\) coherent with the norm topology on \(X\) if it is strongly \(\tau\)-continuous and also has a certain bi-equicontinuity property. In this paper, the author constructs the Hille--Phillips functional calculus (defined in terms of Laplace--Stieltjes integrals) for generators of bi-continuous semigroups. Some rational approximation results known for strongly continuous semigroups are also extended to bi-continuous semigroups. The results provide error estimates for a new class of inversion formulas for the Laplace transform.