Let $U$ and $H$ be Hilbert spaces and $\iota$ be an embedding of $U$ into $H$. Then, $$\pi x:=u\;\;\;\text{for }x\in H\text{ with }x=\iota u+y\text{ for some }u\in U\text{ and }y\in\left(\iota U\right)^\perp$$ is a well-defined mapping $H\to U$. If $\iota$ is an isometry, then $$\langle\iota u,x\rangle_H=\langle\iota u,\iota\left(\pi x\right)\rangle_H=\langle u,\pi x\rangle_U\;\;\;\text{for all }u\in U\text{ and }x\in H\;,$$ i.e. the adjoint of $\iota$ is given by $$\iota^\ast=\pi\;.$$
Can we find a concrete representation of the square-root $Q^{1/2}$ of $Q:=\iota\iota^\ast$?
If that's not possible, can we find it, if $U\subseteq H$ and $\iota$ is the inclusion?