Corollary

Let $u$ be a symmetric partial operator on E. If the partial operator $u$ induces a bijective linear mapping of dom($u$) onto E, then $u$ is self-adjoint.

In fact, the inverse partial operator $u^{-1}$ of $u$ is symmetric with domain E (prop. 9), hence $u^{-1}$ is a self-adjoint element of $\mathscr{L}(E)$ (prop. 10), and $u$ is therefore hermitian (prop. 9).