Lemma 1

The subset Ω(E) is self-adjoint in $\mathscr{L}(E)$.

Let $v\in \Omega (E)$. One has $\|v^*\|=\|v\|\leqslant 1$ therefore the endomorphism $1_E-vv^*$ is positive. It is injective: if $x\in$ Ker(1$_E-vv^*$), one has $vv^*(x) =x$, whence $v^*(v(v^*(x))) =v^*(x)$ then $v^*(x) = 0$, since $1_E-v^*v$ is injective, and finally $x=v(v^*(x)) = 0$. The lemma follows.