 Solution

ID: past-exam-of-the-analysis-course-of-the-university-of-oxford/2011/6/b/iii/solution

⟨ T^{*} x, x ⟩ = ⟨ x, T x ⟩ = \overline{⟨ T x, x ⟩} .

Taking real parts proves

Re ⟨ T^{*} x, x ⟩ = Re ⟨ T x, x ⟩ \geq 0.

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