Two remarks on matrix exponentials (Q1118658)

The author gives some examples to show that in general it is false from \(e^ Ae^ B=e^ Be^ A\) to imply \(AB=BA\) and proves that for two square matrices A and B with algebraic elements it is true. He also proves a perturbation inequality for Hilbert space operators, i.e. if A is normal operator and B is operator on a Hilbert space, then \(\| e^ A-e^ B\| \leq \| e^ A\| (e^{\| A-B\|}-1)\).