From resolvent bounds to semigroup bounds

TL;DR

This paper revisits the Gearhardt-Pr"uss-Hwang-Greiner theorem to provide explicit estimates of semigroup norms based on resolvent bounds, enhancing understanding of semigroup behavior in functional analysis.

Contribution

It offers an explicit estimate of semigroup norms derived from resolvent bounds, clarifying the connection between resolvent bounds and semigroup growth.

Findings

Explicit semigroup norm estimates in terms of resolvent bounds

Simplified proof of the Gearhardt-Pr"uss-Hwang-Greiner theorem

Enhanced understanding of semigroup behavior

Abstract

The purpose of this note 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 the norm of S(t) in terms of bounds on the resolvent of the generator.