Use the norm-attaining factorization $T=AB$ from part v. Then $ST=(SA)B$, parts ii and iv give
$$
\|ST\|_1
\leq\|SA\|_2\|B\|_2
\leq\|S\|\|A\|_2\|B\|_2
=\|S\|\|T\|_1.
$$
Thus the trace-class operators form a left ideal. Applying the result to adjoints gives the corresponding right-ideal estimate as well.

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Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-106/3/b/vi/solution