Proposition 9

Let $v\in \mathscr{L}(F; E)$ be an injective continuous linear mapping of F into E whose image is dense in E. The adjoint of $v$ is an injective continuous linear mapping of E into F and one has $(v^*)^{-1}= (v^{-1})^*$. In particular, if E = F, the endomorphism $v$ is hermitian if and only if the partial operator $v^{-1}$ is self-adjoint.

The partial operator $v^{-1}$ is a closed operator with dense domain from E into F (Example 2 of IV, p. 228), and the adjoint $v^*$ of $v$ is a continuous linear mapping from E into F; it is injective, since the image of $v$ is dense in E (EVT, V, p. 41, prop. 4). Let $s$ (resp. $s'$) be the isometric isomorphism $(x, y)\mapsto (-y, x)$ of $E\oplus F$ onto $F\oplus E$ (resp. the isometric isomorphism $(y, x)\mapsto (-x, y)$ of $F\oplus E$ onto $E\oplus F$), and let $\iota$ (resp. $\iota '$) be the isometric isomorphism $(y, x)\mapsto (x, y)$ of $F\oplus E$ onto $E\oplus F$ (resp. the isometric isomorphism $(x, y)\mapsto (y, x)$ of $E\oplus F$ onto $F\oplus E$). Then $s\circ \iota =-\iota '\circ s'$, whence

by prop. 7 of IV, p. 236. The proposition follows.