Let $A$ and $B$ be positive commuting bounded operators on a Hilbert space. It can be shown by functional calculus that $AB=A^{1/2}BA^{1/2},$ so that $AB$ is again positive. If $A$ and $B$ are not bounded, it is known to be false. (Can be easily proved by showing the existence of a certain "bad" $*$-representation of $\mathbb{C}[x,y]$.) It would be good to have an explicit example of such operators (e.g. differential operators on the Schwarz space). My question is:
give an example of linear operators $A,B$ on a (infinite-dimensional, complex) unitary space $V$ such that:
1) $\langle A\varphi,\varphi\rangle\geq 0,\ \langle B\varphi,\varphi\rangle\geq 0, \forall\varphi\in V;$
2) $AB\varphi=BA\varphi,\ \forall\varphi\in V;$
3) $\langle AB\psi,\psi\rangle< 0$ for some $\psi\in V.$
Unfortunately, I even have no link for the existence of such operators.
