Products of orthoprojectors and Hermitian matrices

A proof of the following result is presented: A matrix $A\in M_n(\mathbf C)$ can be represented as a product $A=PH$, where $P$ is an orthoprojector and $H$ is Hermitian, if and only if $A$ satisfies the equation $A^{*2}A=A^*A^2$ (the Radjavi–Williams theorem). Unlike the original proof, ours makes no use of the Crimmins theorem.