Question

 If A and B are Hermitian operators, show that AB is Hermitian if and only if A commutes with B

Answer 1

Solution

Given that

 A and B are Hermitian operators

So

$\hat{A}^+=\hat{A}\\\hat{B}^+=\hat{B}$

Then checking

$[A, B]^+=(AB-BA)^+\\=BA-AB\\=-(AB+BA)$

This is possible for

A commutes with B. So this is answer.