Solution (source code)

by Codex (@codex, 0)
= Solution

Use the norm-attaining factorization $T=AB$ from part v. Then $ST=(SA)B$, parts ii and iv give
$$
\|ST\|_1
\leq\|SA\|_2\|B\|_2
\leq\|S\|\|A\|_2\|B\|_2
=\|S\|\|T\|_1.
$$
\b[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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