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Convergence estimates for the Magnus expansion II. $C^*$-algebras

Gyula Lakos

TL;DR

Part II develops a spectral framework for Magnus and BCH expansions in $C^*$-algebras and Hilbert space operators, tying convergence to spectral data via the conformal range. It introduces the conformal range as a projection of the Davis-Wielandt shell to bound spectra of time-ordered exponentials and proves absolute Magnus convergence for finite dimensional cases when the cumulative norm is at most $π$, and an extended BCH radius in certain regimes. The work provides extensive 2x2 matrix tools, including exponentials, logarithms, and analytic continuations, and offers a suite of counterexamples that sharpen the $π$ threshold and illustrate limitations in infinite dimensions and unitary-quaternionic settings. It also develops explicit growth estimates for the Magnus expansion and uses hyperbolic geometry models to illuminate spectral behavior within the conformal range.

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 II, we consider the case of $C^*$-algebras, i. e. essentially the case of operators on Hilbert spaces. We present the spectral approach to the Magnus expansion in the context of the conformal range (which is a projection of the Davis--Wielandt shell), allowing a more effective approach. This makes possible to clarify certain convergence properties of the BCH expansion related to the critical cumulative norm $π$. In particular, we prove that for finite dimensional matrices $A,B$, the norm condition $\|A\|_2+\|B\|_2\leqπ$ implies that the BCH expansion of $A$ and $B$ is convergent. Several counterexamples regarding convergence of the Magnus and BCH expansions are presented. In the rest, we prove growth estimates for the Magnus expansion in the setting of Hilbert space operators, both in terms of the overall sum and the individuals terms.

Convergence estimates for the Magnus expansion II. $C^*$-algebras

TL;DR

Part II develops a spectral framework for Magnus and BCH expansions in -algebras and Hilbert space operators, tying convergence to spectral data via the conformal range. It introduces the conformal range as a projection of the Davis-Wielandt shell to bound spectra of time-ordered exponentials and proves absolute Magnus convergence for finite dimensional cases when the cumulative norm is at most , and an extended BCH radius in certain regimes. The work provides extensive 2x2 matrix tools, including exponentials, logarithms, and analytic continuations, and offers a suite of counterexamples that sharpen the threshold and illustrate limitations in infinite dimensions and unitary-quaternionic settings. It also develops explicit growth estimates for the Magnus expansion and uses hyperbolic geometry models to illuminate spectral behavior within the conformal range.

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 II, we consider the case of -algebras, i. e. essentially the case of operators on Hilbert spaces. We present the spectral approach to the Magnus expansion in the context of the conformal range (which is a projection of the Davis--Wielandt shell), allowing a more effective approach. This makes possible to clarify certain convergence properties of the BCH expansion related to the critical cumulative norm . In particular, we prove that for finite dimensional matrices , the norm condition implies that the BCH expansion of and is convergent. Several counterexamples regarding convergence of the Magnus and BCH expansions are presented. In the rest, we prove growth estimates for the Magnus expansion in the setting of Hilbert space operators, both in terms of the overall sum and the individuals terms.

Paper Structure

This paper contains 38 sections, 85 theorems, 637 equations, 18 figures.

Table of Contents

  1. The spectral approach to the Magnus expansion
  2. The conformal range
  3. Conformal range and time-ordered exponentials
  4. Counterexamples
  5. Some restrictions on exponentials
  6. Counterexamples to the Magnus expansion
  7. Counterexamples to the BCH expansion
  8. Exponentials, logarithms, and norms of $2\times2$ matrices
  9. The skew-quaternionic form
  10. Spectral type
  11. Principal and chiral disks
  12. Exponentials
  13. A simple differential equation
  14. The quaternionic alternative
  15. The differential calculus of $\mathop{\mathrm{\slashed{\mathrm C}os}}\nolimits$ and $\mathop{\mathrm{\slashed{\mathrm S}in}}\nolimits$
  16. ...and 23 more sections

Key Result

Theorem 1.1

If $\phi$ is a Banach-algebra $\mathfrak A$ valued ordered measure, and the Magnus expansion absolutely converges to / converges to / Abel-summable to / radially convergent to the value $M$, then

Figures (18)

  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • ...and 13 more figures

Theorems & Definitions (217)

  • Theorem 1.1
  • proof
  • Theorem 1.2
  • proof
  • Theorem 1.3
  • proof
  • Theorem 1.4
  • proof
  • Remark 1.5
  • Lemma 2.1
  • ...and 207 more