Twice applying <Parseval identity for a Hilbertian basis> to the matrix coefficients gives
$$
\|T\|_2^2
=\sum_n\sum_m|\langle Te_n,e_m\rangle|^2
=\sum_m\sum_n|\langle e_n,T^*e_m\rangle|^2
=\|T^*\|_2^2.
$$
Thus $\|T\|_2=\|T^*\|_2$.

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 / i / Solution

 Solution

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