The <operator norm> gives
$$
\|ST\|_2^2
=\sum_n\|STe_n\|^2
\leq\|S\|^2\sum_n\|Te_n\|^2
=\|S\|^2\|T\|_2^2.
$$
Using part i and the first inequality for the adjoints gives
$$
\|TS\|_2=\|S^*T^*\|_2
\leq\|S^*\|\|T^*\|_2
=\|S\|\|T\|_2.
$$

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 / ii / Solution

 Solution

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