 Solution

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

Let

D = T^{*} T - T T^{*}

. This operator is self-adjoint, and

⟨ D x, x ⟩ = ∥ T x ∥^{2} - ∥ T^{*} x ∥^{2} .

If condition (iii) holds, this quadratic form vanishes. For a real inner product, polarization gives

2 ⟨ D x, y ⟩ = ⟨ D (x + y), x + y ⟩ - ⟨ D x, x ⟩ - ⟨ Dy, y ⟩ = 0.

Therefore

D = 0

, which is condition (i). This supplies the nontrivial reverse implication needed for the equivalence.

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