When we have an operator
$$ T ~\colon \mathscr{H} \longrightarrow \mathscr{H} $$
from a Hilbert space to itself, we can use the Riesz representation theorem to prove the existence of the adjoint map $T^*$ with the property
$$ \langle Tx, y \rangle = \langle x, T^* y\rangle \quad \forall x, y \in \mathscr{H} $$
However, if we instead have an operator
$$ S ~\colon \mathscr{H_1} \longrightarrow \mathscr{H}_2 $$
between two different Hilbert spaces $\mathscr{H}_1$ and $ \mathscr{H}_2$, it is often sensible to talk about the map $S^*$ with the property
$$ \langle Sx, y \rangle_{\mathscr{H}_2} = \langle x, S^* y \rangle_{\mathscr{H}_1} \quad \forall x \in \mathscr{H}_1, y \in \mathscr{H}_2 $$ In this case, it appears to me that we cannot use the Riesz representation theorem to prove the existence of $S^*$. This leads me to ask a few questions:
Questions:
- What hypotheses do we need for the operator $S^*$ to exist?
- How would we go about proving this existence?
- Where can I read about adjoints in this more general scenario?
For any continuous linear map $S:\mathscr{H}_1\to\mathscr{H}_2$ between two vector spaces we can define its transpose as $S^*:\mathscr{H}_2^*\to\mathscr{H}_1^*$ by letting $(S^*f)(x)=f(Sx)$ for any $x\in\mathscr{H}_1$ and $f\in\mathscr{H}_2^*$. In order for $S^*f$ to belong to $\mathscr{H}_1^*$, the original map $S$ has to be continuous; otherwise we only get a map between the algebraic duals.
Hilbert spaces are naturally self-dual (by the Riesz representation theorem!), so we can think of $S^*$ as a map $S^*:\mathscr{H}_2\to\mathscr{H}_1$. The dual pairing is (after identifying the spaces with their duals) the inner product, so the property $(S^*f)(x)=f(Sx)$ becomes $\langle S^*f,x\rangle_{\mathscr{H}_1}=\langle f,Sx\rangle_{\mathscr{H}_2}$.
The above discussion proves the existence of $S^*$ assuming continuity. The construction becomes more complicated if $S$ is, say, discontinuous and defined only on a dense subspace. Depending on your taste, things might look more transparent with a concrete operator on concrete spaces, such as the Laplacian on $L^2$, which is discontinuous, densely defined and self-adjoint. In general $S^*$ is continuous if and only if $S$ is.