Improving semi-groups bounds with resolvent estimates

TL;DR

This paper revisits the Gearhardt-Pr"uss-Hwang-Greiner theorem for semigroups, providing explicit resolvent-based estimates on the semigroup norm, with improvements motivated by recent research and optimization considerations.

Contribution

It offers new explicit bounds on semigroup norms in terms of resolvent estimates, enhancing previous results and confirming their optimality.

Findings

Derived explicit semigroup norm estimates from resolvent bounds

Improved upon existing proofs of the Gearhardt-Pr"uss-Hwang-Greiner theorem

Confirmed the optimality of the new bounds through optimization analysis

Abstract

The purpose of this paper is to revisit the proof of the Gearhardt-Pr\"uss-Hwang-Greiner theorem for a semigroup $S(t)$, following the general idea of the proofs that we have seen in the literature and to get an explicit estimate on $\Vert S(t) \Vert$ in terms of bounds on the resolvent of the generator. A first version of this paper was presented by the two authors in ArXiv (2010) together with applications in semi-classical analysis and a part of these results has been published later in two books written by the authors. Our aim is to present new improvements, partially motivated by a paper of D. Wei. On the way we discuss optimization problems confirming the optimality of our results.