 Solution

ID: past-exam-of-the-algebra-analysis-and-differential-equations-2-course-of-the-university-of-oxford/2010/algebra/2/a/ii/solution

⟨ T x, T y ⟩ - ⟨ T^{*} x, T^{*} y ⟩ = ⟨ x, (T^{*} T - T T^{*}) y ⟩ .

If (i) holds, the expression is zero for all

x, y

, giving (ii). Conversely, if this expression vanishes for every

x

, nondegeneracy of the inner product gives

(T^{*} T - T T^{*}) y = 0

for every

y

, so (ii) also implies (i).

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