Non-existence of positive commutators (Q803509)

It is known that when H, A, are two bounded selfadjoint operators, the selfadjoint operator i[H,A] can not be strictly positive. For unbounded H,A this is not true: the classical example being \(Af(x)=xf(x)\) and \(Hf(x)=-if'(x)\) on \(L_ 2(R)\); here \(i[H,A]=I\) on D(H)\(\cap D(A)\). However, under the restriction D(H)\(\subseteq D(A)\) the above statement remains true: Theorem. Let H,A be two selfadjoint Hilbert space operators with D(H)\(\subseteq D(A)\). Then \(i[H,A]\geq aI>0\) is impossible.