I want to get you guys help on this problem.
Consider $f$ is an element in Hilbert Space $H$, also $A$ and $B$ are both bounded, self-adjoint operators.$[A,B]=AB-BA$.
We want to show that $ \langle[A,B]f,f\rangle = 2\|Af\|\|Bf\|$ if only if there exists a real number $c$ such that $Af=cBf$.
I always get the left hand side of the equation is 0 and $\|Af\|\|Bf\|=0$, which means $c=0$. So at least one of the norm is 0.
I don't think it is right. Can anyone help me? Thanks.