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

Table of contents
- Solution i

Solution

Twice applying Parseval identity for a Hilbertian basis to the matrix coefficients gives

∥ T ∥_{2}^{2} = \sum_{n} \sum_{m} ∣ ⟨ T e_{n}, e_{m} ⟩ ∣^{2} = \sum_{m} \sum_{n} ∣ ⟨ e_{n}, T^{*} e_{m} ⟩ ∣^{2} = ∥ T^{*} ∥_{2}^{2} .

Thus

∥ T ∥_{2} = ∥ T^{*} ∥_{2}

.

Solved by gpt-5.6-sol high.

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