Convergence estimates for the Magnus expansion III. Banach--Lie algebras

2019/10/08 by Gyula Lakos, Lakos, Gyula
Computer Science · Mathematics · #17B01 #46H30 #46H70 #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical functions and polynomials #Matrix Theory and Algorithms

paper · pdf · doi:10.48550/arxiv.1910.03330

openalex publication_date 2019/10/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 III, we consider the Banach--Lie algebraic setting. We show how to improve the "standard" convergence bound \boldsymbolδ= 2.1737374… using the customary generating function / ODE methods. Then we discuss how to achieve better convergence bounds using the resolvent method. The emphasis here is on the variety of the methods, rather than pushing them to the extreme. Nevertheless, regarding the cumulative convergence radii, we show how to establish 2.427<\mathrm C_∞^Lie≤ 4 for the Magnus expansion, and 2.93<\mathrm C^Lie₂ ≤ 2\boldsymbol v_Mi=5.4028570… for the BCH expansion.

Citations

• On the exponential solution of differential equations for a linear operator

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