By the definition of the <trace-class operator>[trace norm] and positivity of $|T|$,
$$
\|T\|_1
=\bigl\||T|^{1/2}\bigr\|_2^2
=\sum_n\bigl\||T|^{1/2}e_n\bigr\|^2
=\sum_n\langle|T|e_n,e_n\rangle.
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 106 / 3 / b / iii / Solution

{title2=$\mathcal S_1$}
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A bounded operator $T$ on a separable Hilbert space is trace class when $|T|^{1/2}$ is Hilbert--Schmidt. Its trace norm is
$$
\|T\|_1=\bigl\||T|^{1/2}\bigr\|_{\mathrm{HS}}^2
=\sum_n\langle|T|e_n,e_n\rangle.
$$


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 Incoming links: Trace-class operator