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

We prove a convergence result for the Campbell–Baker–Hausdorff–Dynkin series sum_n Z_n(x,y) in infinite-dimensional Banach–Lie algebras L. In the existing literature, this topic has been investigated when L= Lie(G) is the Lie algebra of a finite-dimensional Lie group G (see [Blanes and Casas, 2004]) or of an infinite-dimensional Banach–Lie group (see [Mérigot, 1974]). Indeed, one can obtain a suitable ODE for gamma(t) = sum_n Z_n(x,ty), which follows from the well-behaved formulas for the differential of the Exponential Map of the Lie group G. 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 s,t over the free unital associative algebra generated by two non-commuting indeterminates x,y.