SCHWEITZER 2001 Problem 8

Let $H$ be a complex Hilbert space. The bounded linear operator $A$ is called positive if $\langle Ax, x\rangle \geq 0$ for all $x\in H$ . Let $\sqrt A$ be the positive square root of $A$ , i.e. the uniquely determined positive operator satisfying $(\sqrt{A})^2=A$ . On the set of positive operators we introduce the $$A\circ B=\sqrt A B\sqrt B$$ operation. Prove that for a given pair $A, B$ of positive operators the identity $$(A\circ B)\circ C=A\circ (B\circ C)$$ holds for all positive operator $C$ if and only if $AB=BA$ .