Proposition 12

Let $E_1$ and $E_2$ be complex Hilbert spaces and $u\in \mathscr{L}(E_1; E_2)$. Let $(j,|u|)$ be the polar decomposition of $u$. For $u$ to be bijective, it is necessary and sufficient that $|u|$ be invertible in $\mathscr{L}(E_1)$ and that $j$ be an isomorphism of $E_1$ onto $E_2$.

The condition is sufficient. Conversely, if $u$ is bijective, then $u^*u$ is invertible in $\mathscr{L}(E_1)$, and $|u|= (u^*u)^{1/2}$ is also invertible since its spectrum is contained in $\mathbf{R}_+^*$. Moreover, Ker($j$) $=$ Ker($u$) $=\{0\}$ and Im($j$) $=$ Im($u$) $= F$, whence $j$ maps $E_1$ isometrically onto $E_2$.