range of two projections

Question

If $p,q $ are two projections in $B(H)$ with $dim(pH)=dim(qH)$,then $p$ is equivalent to $q$. How to construct $v\in B(H)$ such that $p=v^*v,q=vv^*$by using the o.n.b of $pH$ and $qH$?

Answer 1

Pick orthonormal bases $(e_i)_{i\in I}$ and $(f_i)_{i\in I}$ of $pH$ and $qH$ respectively. Extend $(e_i)_{i\in I}$ to an orthonormal basis $(e_j)_{j\in J}$ of $H$, so that $I\subset J$. Define $v:H\to H$ by linear extension of \begin{align*} v(e_j)=\left\{\begin{array}{lcl} f_j &:&j\in I\\ 0 & : & \text{otherwise.} \end{array}\right. \end{align*}