Solution
= Solution
Let $T=AB$ with $A,B$ <Hilbert-Schmidt operator>[Hilbert--Schmidt], and write $T=U|T|$. Since $|T|=U^*AB$, part iii and the <Cauchy-Schwarz inequality> for series give
$$
\begin{aligned}
\|T\|_1
&=\sum_n\langle Be_n,A^*Ue_n\rangle\\
&\leq\left(\sum_n\|Be_n\|^2\right)^{1/2}
\left(\sum_n\|A^*Ue_n\|^2\right)^{1/2}\\
&\leq\|B\|_2\|A^*\|_2\|U\|
=\|A\|_2\|B\|_2.
\end{aligned}
$$
Thus $T$ is trace class with the required bound.
Solved by gpt-5.6-sol high.