There are a lot of questions here about square root operators, but none of them addresses the basic question of existence, and I didn't find a very beefy section in Wikipedia talking about this, so I'll ask it here. Let $A$ be a bounded positive-semidefinite Hermitian operator on Hilbert space. The claim is that there exists a positive-semidefinite Hermitian operator $B$ such that $B\circ B=A$. I've got no idea how you would prove this. Any tips or references would be appreciated.