Convergence proof for Goldberg's exponential series (Q1123991)

In 1956, \textit{K. Goldberg} obtained the expansion \[ z=x+y+xy-yx+... \] for the element z such that \(e^ z=e^ xe^ y\). Here all computations are made in the completed free associative algebra generated by two noncommuting elements x and y; thus z is a (formal) infinite sum of words in x, y, and Goldberg gave an explicit method to compute their coefficients [Duke Math. J. 23, 13-21 (1956; Zbl 0070.252)]. Relying on this method, the author proves here two convergence results for the Goldberg series. 1) If x, y belong to an associative normed algebra (that is \(\| ab\| \leq \| a\| \| b\|)\), then the series converges for \(\| x\| <1\), \(\| y\| <1\). Besides, it is known that z can also be written as a corresponding series of Lie brackets \(z=x+y+[x,y]+... \). The second result is: 2) If x, y belong to a normed Lie algebra (that is \(\| [a,b]\| \leq \| a\| \| b\|)\), then the commutator series converges for \(\| x\| <1\), \(\| y\| <1\).