What is the Hermitian adjoint of operator $\hat{K}\psi = \psi^*$? For many operators, their adjoint can be expressed as a function of other known operators, for example $$\hat{T}_a^\dagger = \hat{T}_{-a} \\ \hat{p}_x^\dagger = \hat{p}_x$$ where $\hat{T}_a \psi (x) = \psi (x+a)$ and $\hat{p}_x \psi(x)= -ih \frac{\partial \psi(x)}{\partial x}$.
But if we consider the operator $\hat{K}\psi (x) = \psi^* (x)$ (here $^*$ denotes complex conjugate) then, from definition,
$$ \langle \phi |\hat{K}^\dagger |\psi \rangle =  \langle \psi |\hat{K} |\phi \rangle^* =  \langle \psi |\phi^* \rangle^* = \langle \psi^* |\phi \rangle   $$
Can it be expressed as $\langle \phi |\hat{A}|\psi \rangle$ for some operator $\hat{A}$? If not, why?
Also, what does this result imply? For example, is $\hat{K}$ Hermitian?
$$ \langle \phi |\hat{K}^\dagger |\psi \rangle = \langle \phi |\hat{K} |\psi \rangle \\ \therefore \langle \psi^* |\phi \rangle = \langle \phi |\psi^* \rangle$$
So it is Hermitian only for certain wavefunctions?
 A: The operator $K$ on the Hilbert space $H$ is antilinear and satisfies
$$(\eta,K^*\psi)=\overline{(\eta,K\psi)},\qquad\forall \eta,\psi\in H,$$
which is not what you would expect for self-adjointness.
You can consider the conjugate Hilbert space $\overline H$ and turn $T$ into a linear map by regarding it as $T:H\to\overline H$. If you then take
$$(\eta,\psi)_{\overline H}:=\overline{(\eta,\psi)}_H$$
as the inner product on $\overline H$, the above identity becomes
$$(\eta,K^*\psi)_{\overline H}=(\eta,K\psi)_H,\qquad\forall \eta,\psi\in H,$$
but since the operator is now between two "different" Hilbert spaces, this doesn't imply that $K$ is self-adjoint.
A: $\hat{K} \Psi = \Psi^*$ is not a well defined operator at all. On the one hand it is clear that $\hat{K^2} = 1$ and $\hat{K}$ is therefore real. On the other hand by your definition
$$
<\Psi| K |\Psi> = \int \Psi^{*2} d^3r \not\in \mathbb{R}
$$
It is ill defined because it does not map from the space of the Kets into the space of the kets but instead from Ket to Bra space.
