Prove that the product of two positive linear operators is positive if and only if they commute.
Solution
Let \(S\) and \(T\) be positive operators.
1. If \(ST\) is positive, then \(S\) and \(T\) commute.
Since \(S\) and \(T\) are positive, they are self-adjoint. If \(ST\) is positive, then \(ST\) is also self-adjoint. Therefore \[ ST=(ST)^*=T^*S^*=TS. \] Hence \(S\) and \(T\) commute.
2. If \(S\) and \(T\) commute, then \(ST\) is positive.
Because \(S\) and \(T\) are commuting self-adjoint operators, they can be simultaneously diagonalized by an orthonormal basis. Thus in some orthonormal basis, \[ [S]=\operatorname{diag}(s_1,\ldots,s_n), \qquad [T]=\operatorname{diag}(t_1,\ldots,t_n), \] with every \(s_i>0\) and every \(t_i>0\). Hence \[ [ST]=\operatorname{diag}(s_1t_1,\ldots,s_nt_n), \] whose diagonal entries are all positive. Therefore \(ST\) is positive.
Thus \[ \boxed{ST\text{ is positive}\iff ST=TS.} \]