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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?
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For any 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 $f\in\mathscr{H}_2^*$. 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 $S^*$. The construction above does not rely on $S$ for being continuous, so $S^*$ exists with the same properties for any linear $S$; one can show that $S^*$ is continuous if and only if $S$ is.

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