Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2010 / Algebra / 2 / a / i

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- Solution 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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