Adjoining Commutes with Inverting

Theorem
Let $H$ be a Hilbert space.

Let $A \in B \left({H}\right)$ be a bounded linear operator.

Let $A^{-1} \in B \left({H}\right)$ be an inverse for $A$.

Then the adjoint of $A$, $A^*$, is invertible.

Furthermore, $\left({A^*}\right)^{-1} = \left({A^{-1}}\right)^*$.