Convergence estimates for the Magnus expansion I. Banach algebras

TL;DR

This paper reviews and simplifies convergence estimates for the Magnus and Baker--Campbell--Hausdorff expansions within Banach algebras, establishing precise convergence radii and developing the resolvent method in the analytic setting.

Contribution

It provides improved convergence bounds for the Magnus and BCH expansions and introduces the resolvent method in the Banach algebra context.

Findings

Magnus expansion convergence radius is 2

Baker--Campbell--Hausdorff series convergence radius is approximately 2.898

Develops the resolvent method for analytic Banach algebras

Abstract

We review and provide simplified proofs related to the Magnus expansion, and improve convergence estimates. Observations and improvements concerning the Baker--Campbell--Hausdorff expansion are also made. In this Part I, we consider the general Banach algebraic setting. We show that the (cumulative) convergence radius of the Magnus expansion is $2$; and of the Baker--Campbell--Hausdorff series is $\mathrm C_2=2.89847930\ldots$. More generally, the resolvent method is developed in the analytic setting.