A operator is unitary if and only if it is a surjective isometry

Let $U$ be a unitary, then $UU^*=U^*U=1$.

$U^*U=1$ implies that $U$ is an isometry. Also $UU^*=1 $ implies that $U$ is onto, because ($UU^*H=H$).

Conversely, Suppose $U$ is an onto isometry, so $U^*U=1$. It's just necessary to show that $UU^* = 1$. If $\eta\in H$, then there is $\xi\in H$ such that $U\xi=\eta$ ($U $ is onto). Also $U$ is an isometry, so $ \xi=U^*U\xi = U^*\eta$ .Now clearly $UU^*\eta = U\xi =\eta$ for $\eta \in H$. Therefore $UU^*=1$.