Lie product type formulas for the logarithm

Abstract

In this paper, we prove various Lie product type formulas for the logarithm. A sample result: Let \(k\ge 2\) be a natural number, \(X_{1}\),..., \(X_{k}\), Y Banach algebras with unit and \(T:X_{1}\times \cdot \cdot \cdot \times X_{k}\rightarrow Y\) a continuous k-linear operator such that \(T\left( \textbf{1},...,\textbf{1}\right) =\textbf{1}\). Let \(\left( a_{n}\right) _{n\in \mathbb {N}}\) be a sequence of natural numbers with \( \lim _{n\rightarrow \infty }a_{n}=\infty \). Then for all \(\left( x_{1},...,x_{k}\right) \in X_{1}\times \cdot \cdot \cdot \times X_{k}\) we have

$$\begin{aligned} & \lim \limits _{n\rightarrow \infty }\left[ T\left( \textbf{1}+\ln \left( \textbf{1}+\frac{x_{1}}{a_{n}}\right) ,...,\textbf{1}+\ln \left( \textbf{1}+ \frac{x_{k}}{a_{n}}\right) \right) \right] ^{a_{n}} \\ & \quad =e^{T\left( x_{1},\textbf{1},...,\textbf{1}\right) +T\left( \textbf{1},x_{2},\textbf{1},...,\textbf{1}\right) +\cdot \cdot \cdot +T\left( \textbf{1},...,\textbf{1},x_{k}\right) }. \end{aligned}$$