A commutativity theorem for semibounded operators in Hilbert space

Abstract:

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.