Given a linear operator $A: \mathbb{C} \rightarrow \mathbb{R}$, and scalars $x \in \mathbb{C}$, $y \in \mathbb{R}$, I think we should have
\begin{equation}
\langle Ax,y\rangle = \langle x,A^*y\rangle,
\end{equation}
where $A^*: \mathbb{R} \rightarrow \mathbb{C}$ is the adjoint of $A$.
However, as a concrete example, take $A$ to be the operator that returns the real part of its argument. In this case, I think an adjoint would be $A^*y = y + 0i$. Inserting into the equation above, we get
\begin{equation}
\langle \Re(x), y\rangle = \langle x, y + 0i\rangle,
\end{equation}
or: $\Re(x)y$ = $xy$, which is not true. Where is my mistake?
Best Answer
My mistake (thanks to s.harp for pointing it out in a comment) is that complex-to-real operators, such as the example $Ax = \Re{x}$ that I gave, are $\mathbb{R}$-linear, not $\mathbb{C}$-linear. They satisfy the condition to be linear operators, $A(ax+by) = aA(x) + bA(y)$, only when $a$ and $b$ are real. This apparently implies that we need to perform the inner product differently, so $\langle x, y + 0i \rangle = \Re{x}\cdot y + \Im{x}\cdot0 = \Re{x}\cdot y$, satisfying the desired equality in my question.