Convergence estimates for the Magnus expansion I. 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.