Let $\mathbb H$ be a Hilbert space and let $\mathcal B(\mathbb H)$ be the bounded operators on $\mathbb H$. Let $J,K\in \mathcal B(\mathbb H)$ such that $ J=J^*, K=-K^*. $ Then the commutator $[J,K]$ is selfadjoint, equal to $JK+(JK)^*$.

**Claim.** If $[J,K]\ge 0$, then $[J,K]=0$.

**Question.** Is it true, is it well-known? What I know is that it is impossible to have $[J,K]=I$, showing that Quantum Mechanics needs unbounded operators to get the Uncertainty Principle, based upon the non-commutation of operators such as $D_x=\frac1{i}\frac{d}{dx}=J$ and $ix=K$. Here, we have a stronger statement, claiming that it is not even possible that the spectrum of the self-adjoint bounded operator $[J,K]$ is included in the non-negative half-line. (I guess that the above Claim also prevents the spectrum to be included in the non-positive half-line.)