 Solution

ID: past-exam-of-the-analysis-course-of-the-university-of-oxford/2010/b/5/v/solution

For

x \in ker (S + T)

0 = ⟨(S + T) x, x ⟩ = ⟨ S x, x ⟩ + ⟨ T x, x ⟩ .

Both terms are nonnegative because

S, T

are positive operators. Their sum is zero, so each vanishes. In particular,

⟨ T x, x ⟩ = 0.

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