Do antilinear maps have adjoints? Let $T: H \to H$ be a continuous antilinear map on a Hilbert space $H$, that is $T(\alpha \xi + \eta)= \overline{\alpha} T \xi + T \eta$ for $\xi, \eta \in H$ and $\alpha \in \mathbb{C}$. Does there exist an antilinear map $T^*: H \to H$ such that
$$\langle T^* \xi, \eta\rangle = \langle \xi, T\eta\rangle$$
for all $\xi, \eta \in H$ (or a variation on this).
I could try to mimique the "usual" proof for bounded linear maps, but maybe there is a smart way to avoid this kind of work by reducing to this case?
 A: Well, $$\langle T^*\xi,\eta\rangle=\langle\xi, T\eta\rangle$$
cannot stand if $T,T^*$ are antilinear: the LHS is anti-linear on $\xi$ while RHS is linear on $\xi$. But if written as
$$\langle\eta,T^*\xi\rangle=\langle\xi,T\eta\rangle$$
then yeah, check this out:
Forget about scalar multiplication and inner product on $H$ for a bit and define new ones as $\lambda* x:=\bar{\lambda}\cdot x$ where in the RHS  the scalar multiplication is the old one. Also define $\langle x,y\rangle_2:=\langle y,x\rangle_H$ where the RHS inner product is the old one and note that $H$ endowed with the new scalar multiplication and $\langle-,-\rangle_2$ is a new Hilbert space. Also, $T$ is a bounded linear operator when seen acting from $H$ with the old structure towards $H$ with the new structure, thus it has an adjoint operator $T^*$ which acts from $H$ with the new structure to $H$ with the old structure, so
$$\langle Tx,y\rangle_2=\langle x,T^*y\rangle_H$$
i.e.
$$\langle y,Tx\rangle_H=\langle x,T^*y\rangle_H $$
which is what you want. Since $T^*$ is a linear operator for the new scalar multiplication, $T^*$ is antilinear for the original one.
