On the convergence of the Campbell–Baker–Hausdorff–Dynkin series in infinite-dimensional Banach–Lie algebras

Abstract

We prove a convergence result for the Campbell–Baker–Hausdorff–Dynkin series in infinite-dimensional Banach–Lie algebras . In the existing literature, this topic has been investigated when is the Lie algebra of a finite-dimensional Lie group (see [Blanes and Casas, 2004]) or of an infinite-dimensional Banach–Lie group (see [Mérigot, 1974]). Indeed, one can obtain a suitable ODE for , which follows from the well-behaved formulas for the differential of the Exponential Map of the Lie group . The novelty of our approach is to derive this ODE in any infinite-dimensional Banach–Lie algebra, not necessarily associated to a Lie group, as a consequence of an analogous abstract ODE first obtained in the most natural algebraic setting: that of the formal power series in two commuting indeterminates over the free unital associative algebra generated by two non-commuting indeterminates .

Keywords:

• Banach–Lie algebra
• Campbell–Baker–Hausdorff–Dynkin Theorem
• domain of convergence
• formal power series

AMS Subject Classification:

• Primary: 17B01
• 17B65
• Secondary: 17B05
• 46H99

Acknowledgement

We would like to thank Professor Jean Michel for making the manuscript [5] available to us.

Notes

1 We say that a power series is deduced from a power series , if is the Taylor series of the function about some point belonging to the disc of convergence of .

2 An analysis of the contents of the unpublished manuscript [5] is available in the preprint [30] available to us.