When the inverse of bounded linear operator equal to its adjoint operator in Hilbert $(A^{-1} = A^*)$
Solution 1:
Such an operator is called Unitary. A sufficient condition is that
- $A$ is surjective (in fact dense image is enough)
- $A$ preserves the inner product on $H$: $$\langle Ax, Ay\rangle = \langle x,y\rangle$$
Many examples come from the following situation. Let $B:H\to H$ be a self adjoint operator on $H$, that is $B^*=B$ and $Dom(B)=Dom(B^*)$, and define
$$A = e^{iB}\ .$$
Then $A$ is unitary.