equivalent projections

Question

Suppose $p,q$ are two equivalent projections in $B(H)$,do $p(H)$ and $q(H)$ have the same dimension?

Answer 1

Yes, if $p=v^*v$ and $q=vv^*$, then $v$ is an isometry from $pH$ onto $qH$.

You have $$ vp\xi=vv^*v\xi=qv\xi\in qH. $$ Also, $$ q\xi=q^2\xi=vv^*vv^*\xi=vpv^*xi\in vpH, $$ so $v$ is onto $qH$. And $$ \|vp\xi\|^2=\langle vp\xi,vp\xi\rangle=\langle v^*vp\xi,p\xi\rangle=\langle p^2\xi,p\xi\rangle=\langle p\xi,p\xi\rangle=\|p\xi\|. $$