Lemma 2.1.4.label Let $H$ be a Hilbert space and $T: D(T) \to T(H)$ be an injective, self-adjoint operator with dense range, then $T^{-1}: T(H) \to D(T)$ is also self-adjoint.
Proof. Let $y \in D((T^{-1})^{*})$, then there exists $\phi \in H$ such that $\dpb{T^{-1}x, y}H = \dpb{x, \phi}H$ for all $x \in T(H)$. Under the substitution $x = Tz$, this implies that $\dpb{z, y}H = \dpb{Tz, \phi}H$ for all $x \in T(H)$. Hence $y = T^{*}(\phi) = T(\phi) \in T(H)$.$\square$