Is the adjoint operator defined for any two inner product spaces with possibly different scalar fields?
Here is a confusing example. Assume that $H_{1}$ is an inner product space defined on the set of hermitian matrices over the real vector space. And $H_{2}$ is an inner product space defined on the set of complex matrices over the complex vector space.
Let $T:H^{n} \rightarrow C^{n}$ be a linear operator from the hermitian matrices to complex matrices. Then its adjoint is the unique $T^{*}$ such that $$\langle TX,Y\rangle_{H_{2}} = \langle X, T^{*}Y\rangle_{H_{1}}$$
But $\langle TX,Y\rangle_{H_{2}}$ is a complex number, and $\langle X, T^{*}Y\rangle_{H_{1}}$ is a real number. How could they be equal for any $X$ and $Y$?