I am reading Vladimir Arnold's book on ordinary differential equations, and there is a problem on the Sample Examination Problems' section that is giving me a lot of troubles. The question is:
Can the operators $A$ and $B$ fail to commute if $e^{A}=e^{B}=e^{A+B}= E$ , where $E$ is the identity.
So far I can deduce that the eigenvalues of $A$ and $B$ most be imaginary numbers of the form $i 2\pi k$, where $k$ in an integer. Because if we write $A$ in its canonical form $A=Q^{-1}\Delta Q$, where $\Delta$ is the canonical form of $A$, using the properties of exponentials $e^{A}=e^{Q^{-1}\Delta Q}= Q^{-1} e^{\Delta} Q=E$. Then $e^{\Delta}=Q (Q^{-1} e^{\Delta} Q)Q^{-1}=Q (E) Q^{-1}=E$. A similar conclusion holds for $B$ and $A+B$.
Since $\Delta$ is the canonical form $e^{t\Delta}$ it has polynomials $\frac{t^{m}}{m!}$ outside the diagonal if there is a Jordan-Block, but by evaluating on $t=1$ we find that they must be zero, so the matix $\Delta$ must be a diagonal one. And using the complex exponential we conclude that the eigenvalues of $A$ are of the form $i 2\pi k$.
And that is as far as I go, I have tried some matrices but they all appear to commute. I try using Zassenhaus' formula, but no luck.
Thanks.
