On M. M\'erigot's theorem on the convergence domain of the Campbell-Baker-Hausdorff-Dynkin series

TL;DR

This paper discusses Michel Mérigot's 1974 proof that extends the convergence domain of the Campbell-Baker-Hausdorff-Dynkin series within Banach-Lie algebras, based on a novel ODE solution lifetime theorem.

Contribution

It presents a detailed proof of an enlarged convergence domain for the series, utilizing a new theorem on the lifetime of solutions to certain Cauchy problems.

Findings

Extended convergence domain of the series proven.

Detailed proof based on a new ODE solution lifetime theorem.

All technical details provided for the proof.

Abstract

The aim of this manuscript is to present the proof given by Michel M\'erigot in 1974 for an enlarged convergence domain of the Campbell-Baker-Hausdorff-Dynkin series in the Lie algebra of a Banach-Lie group. This proof is based on a theorem, of independent interest, on the lifetime of the solution of a Cauchy problem. We furnish all the details for this ODE result in Appendix A.