Relations between projections $p,q,r$ in $B(H)$

Question

Let $p,q,r$ be three projections in $B(H)$ for some Hilbert space $H$. Suppose that there exists a partial isometry $u \in B(H)$ such that $q=u^*u$ and $r=uu^*$. Also suppose that $p\le q$. Now I want to show that, there exists an element $v \in B(H)$ such that $p=vv^*$ and $v^*v\le r.$ Please help me to solve this. Any comments and suggestion will be appreciated. Thank you.

Answer 1

The solution is direct: just take $v=pu^*$. Then, $$v^*v=upu^*\le uu^*=r$$ and $$vv^*=pu^*up=pqp=p$$ where we used that when $p\le q$ for two projections, then $pq=qp=q$ (exercise).