If $A^{1/2}B^{1/2}=0$. Why $BA=0\;?$

Question

Let $E$ be an infinite-dimensional complex Hilbert space.

Let $A,B\in \mathcal{L}(E)^+$ be such that $A^{1/2}B^{1/2}=0$. Why $$BA=0\;?$$

Answer 1

Multiplying by $A^{1/2}$ on the left and by $B^{1/2}$ on the right you get $AB=0$. Then $$ BA=B^*A^*=(AB)^*=0. $$