A commutativity theorem for semibounded operators in hilbert space

Let $A$ and $B$ be semibounded (bounded from below) operators in a Hilbert space $\mathfrak H$ and $\mathfrak D$ a dense linear manifold contained in the domains of $AB$, $BA$, $A^2$, and $B^2$, and such that $ABx=BAx$ for all $x$ in $\mathfrak D$. It is shown that if the restriction of $(A+B)^2$ to $\mathfrak D$ is essentially self-adjoint, then $A$ and $B$ are essentially self-adjoint and $\bar A$ and $\bar B$ commute, i.e. their spectral projections permute.