Suppose $A$ is a Hilbert-Schmidt operator on a Hilbert space $\mathcal{H}$, then when is it true that
$$\langle Ax, x \rangle_{\mathcal{H}} = \langle A, x \otimes x \rangle_{\text{HS}}, \quad \forall \, x \in \mathcal{H}$$
I know from the definition of Hilbert-Schmidt inner product and the definition of tensor product that
$$
\langle A, x \otimes x \rangle_{\text{HS}} = \sum_{j \in J} \langle A e_j, \langle x, e_j \rangle_{\mathcal{H}} x \rangle_{\mathcal{H}}
$$
where $\{e_j\}_{j \in J}$ is an arbitrary ONB of $\mathcal{H}$. However, I am unable to simplify further. Any help would be appreciated.
If the above result is incorrect, in general, I am interested in going from an expression of the form $\langle Ax, x \rangle_{\mathcal{H}}$ to an expression of the form $\langle A, x \otimes x \rangle_{\text{HS}}$. How could I do that?
Also, I would really appreciate some references where I can look into these kind of results in more detail.
Best Answer
What you need to do is use an orthonormal basis whose first element is $x/\|x\|$. Then $$ \langle A,x\otimes x\rangle_{\rm HS}=\operatorname{Tr}(A(x\otimes x))=\langle A(x\otimes x)\tfrac{x}{\|x\|},\tfrac{x}{\|x\|}\rangle=\langle Ax,x\rangle, $$ since $(x\otimes x)x=\|x\|^2\,x$.