Suppose $H$ is a Hilbert space. Let $U:H\rightarrow H$ be a bounded linear operator.
Does $U^{*}U$ is a projection implies $UU^{*}$ is a projection?
Of course, once we prove this, it follows that one of them is a projection is equivalent to another one is a projection.
Yes. We have the following equivalences:
(a) $U^*U$ is a projection,
(b) $U^*UU^*=U^*$,
(c) $UU^*U=U$,
(d) $UU^*$ is a projection.
The proofs that (b) implies (a), (c) implies (d), and (b) and (c) are equivalent, are trivial. To show (a) implies (c), by the $C^*$-identity we have \begin{align*} \|UU^*U-U\|^2&=\|(U^*UU^*-U^*)(UU^*U-U)\|\\ &=\|U^*UU^*UU^*U-2U^*UU^*U+U^*U\|\\ &=0. \end{align*} Then proving that (d) implies (b) is completely analogous.
