Is the composition of positive, self-adjoint operators on a Hilbert space positive?

Question

Let $A,B$ be self-adjoint, bounded, linear operators on a Hilbert space $H$ over $\mathbb{R}$ such that $\langle Ax,x\rangle\geq 0$ and $\langle Bx,x\rangle\geq 0$ for all $x\in H$. Does it also hold that $\langle ABx,x\rangle\geq 0$ for all $x\in H$?

Answer 1

Assume that $A,B$ are positive operators. A positive operator is self-adjoint. So, if $AB \ge 0$, then it is necessary that $(AB)^*=AB$, or $BA=AB$. In that case $AB \ge 0$ because, you may write $A=\sqrt{A}^2$ where $\sqrt{A}$ is the unique positive square root of $A$ that commutes with everything that commutes with $A$. Then you end up with $\sqrt{A}B=B\sqrt{A}$ and $$ AB=\sqrt{A}\sqrt{A}B=\sqrt{A}B\sqrt{A} \ge 0. $$