Convergence estimates for the Magnus expansion IE. Finite dimensional Banach algebras

TL;DR

This paper improves convergence estimates for the Magnus expansion in finite-dimensional Banach algebras, providing simplified proofs and counterexamples for the Baker--Campbell--Hausdorff expansion, thus advancing understanding of their convergence properties.

Contribution

It offers improved convergence bounds for the Magnus expansion and presents concrete counterexamples for the Baker--Campbell--Hausdorff expansion in finite-dimensional Banach algebras.

Findings

Magnus expansion converges if the cumulative norm is less than 2+ε.

Counterexamples exist for cumulative norms greater than 2.

Simplified proofs enhance understanding of convergence criteria.

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 IE, we consider the case of finite dimensional Banach algebras. We show that Magnus expansion is convergent (and works in logarithmic sense) if the cumulative norm $< 2+\varepsilon_{n}$, where $\varepsilon_{n}$ is a positive number depending on the dimension $n$ of the Banach algebra. We also show concrete finite-dimensional counterexamples of multiple Baker--Campbell--Hausdorff type for any cumulative norm $>2$ (necessarily of possibly great dimension).