A condition to ensure that adjoint and inverse commute

Question

Let $A$ be a densely defined operator on a Hilbert space $H$. Let us assume that $A$ is injective and $Ran(A)$ is dense. Then I want to show that $$ (A^*)^{-1} = (A^{-1})^*. $$

So far, I was able to show that $A^*$ is also densely defined operator and it is injective and $Ran(A^*) $ is dense. But I can't go any further. Any help will be very much appreciated!

Answer 1

Hint: Just write out the definitions of $x=(A^*)^{-1}y$ and $x=(A^{-1})^*y$, to find that they are equivalent conditions.

The full details are hidden below.

Note that $x=(A^*)^{-1}y$ is equivalent to $A^*x=y$ which is equivalent to $\langle y, z\rangle=\langle x, Az\rangle$ for all $z$ in the domain of $A$.

On the other hand, $x=(A^{-1})^*y$ is equivalent to $\langle x,w\rangle=\langle y, A^{-1}w\rangle$ for all $z$ in the domain of $A^{-1}$. The domain of $A^{-1}$ is just the set of $w$ that have the form $Az$ for some $z$ in the domain of $A$, so this condition is equivalent to $\langle x,Az\rangle=\langle y,z\rangle$ for all $z$ in the domain of $A$.