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Convergence estimates for the Magnus expansion IA. Uniformly convex algebras

Gyula Lakos

TL;DR

This work demonstrates that uniform convexity in Banach algebras strengthens convergence properties of the Magnus expansion beyond the classical Banach-algebra setting. By developing and deploying resolvent-based methods (including the delay method and chronological decomposition) and resolvent/estimating kernels, the authors obtain explicit, improved convergence radii under the UM D_q and UM Q_q frameworks, with precise results in the Hilbert-space case showing radii exceeding 2.0408... and approaching π in idealized limits. The construction of universal and quasi-free algebras provides a robust model for worst-case behavior and enables practical, computable bounds via linear programming and kernel estimates. The results also extend to BCH/Baker–Campbell–Hausdorff expansions, clarifying how permutation-type inequalities influence convergence and offering a pathway to extensions into Banach–Lie algebras (to be addressed in Part III). Overall, the work offers a systematic, computable framework for assessing Magnus/BCH convergence in broad algebras, with concrete numerical bounds and methods that can guide both theory and applications.

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 IA, we consider uniform convexity. Notions of uniformly convex algebras are discussed, and uniform convexity is shown to improve convergence estimates.

Convergence estimates for the Magnus expansion IA. Uniformly convex algebras

TL;DR

This work demonstrates that uniform convexity in Banach algebras strengthens convergence properties of the Magnus expansion beyond the classical Banach-algebra setting. By developing and deploying resolvent-based methods (including the delay method and chronological decomposition) and resolvent/estimating kernels, the authors obtain explicit, improved convergence radii under the UM D_q and UM Q_q frameworks, with precise results in the Hilbert-space case showing radii exceeding 2.0408... and approaching π in idealized limits. The construction of universal and quasi-free algebras provides a robust model for worst-case behavior and enables practical, computable bounds via linear programming and kernel estimates. The results also extend to BCH/Baker–Campbell–Hausdorff expansions, clarifying how permutation-type inequalities influence convergence and offering a pathway to extensions into Banach–Lie algebras (to be addressed in Part III). Overall, the work offers a systematic, computable framework for assessing Magnus/BCH convergence in broad algebras, with concrete numerical bounds and methods that can guide both theory and applications.

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 IA, we consider uniform convexity. Notions of uniformly convex algebras are discussed, and uniform convexity is shown to improve convergence estimates.
Paper Structure (27 sections, 81 theorems, 313 equations)

This paper contains 27 sections, 81 theorems, 313 equations.

Table of Contents

  1. Uniform convexity in Banach algebras
  2. Uniform convexity --- definitions
  3. On the variants of uniform convexity
  4. Universal algebras and the convergence problem
  5. Some special algebras
  6. Certain more general algebras
  7. Selection estimates
  8. The basics of the resolvent approach and the delay method
  9. The principles of the resolvent approach
  10. Resolvent estimates via Euler's recursion and the delay method
  11. The chronological decomposition method
  12. The plain method.
  13. The spectrally improved method.
  14. The resolvent kernel method
  15. The resolvent generating kernels
  16. ...and 12 more sections

Key Result

Theorem 1.5

Consider the Banach space $L^p(\mu)$ where $1<p<+\infty$. Let us denote the norm by $|\cdot|$. Let $q=\max(p,\frac{p}{p-1})$ and $q'=\min(p,\frac{p}{p-1})$ . Let $r$ be such that $q\leq r <+\infty$, and $r'=\frac{r}{r-1}$, thus $1<r'\leq q'$. Then (This is actually the special case "$s=r'$" of Boas' inequality.) ∎

Theorems & Definitions (183)

  • Definition 1.1
  • proof : Remark
  • Definition 1.2
  • proof : Remark
  • Definition 1.3
  • proof : Remark
  • Definition 1.4
  • proof : Remark
  • Theorem 1.5: Boas Bo (1940)
  • Theorem 1.6: Clarkson C1 (1936)
  • ...and 173 more