Inequalities for exponentials in Banach algebras

For commuting elements x, y of a unital Banach algebra ℬ it is clear that ${\displaystyle \parallel e^{x+y}\parallel \le \parallel e^x\parallel \parallel e^y\parallel }$. On the order hand, M. Taylor has shown that this inequality remains valid for a self-adjoint operator x and a skew-adjoint operator y, without the assumption that they commute. In this paper we obtain similar inequalities under conditions that lie between these extremes. The inequalities are used to deduce growth estimates of the form ${\displaystyle \parallel e{\prime }^{}\parallel \le c(1+|\xi |{⟩}^s}$ for all ${\displaystyle \xi \in R^m}$, where ${\displaystyle x=(x_1,...,x_m)\in {ℬ}^m}$ and c, s are constants.