Adjoining Commutes with Inverting

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Theorem

Let $H$ be a Hilbert space.

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

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

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

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

By definition of inverse, one has $AA^{-1} = I_K$, where $I_K$ is the identity operator on $K$.

Now observe from Adjoint of Composition that:

$I_K = I_K^* = \left({AA^{-1}}\right)^* = \left({A^{-1}}\right)^*A^*$.

Similarly, one has:

$I_H = I_H^* = \left({A^{-1}A}\right)^* = A^*\left({A^{-1}}\right)^*$

Hence, by definition of inverse, $\left({A^*}\right)^{-1} = \left({A^{-1}}\right)^*$.

This also means that $A^*$ is invertible.

$\blacksquare$