 Solution

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

A separate expansion gives

∥ (I + T) x ∥^{2} = ∥ x ∥^{2} + ∥ T x ∥^{2} + 2 Re ⟨ T x, x ⟩ \geq ∥ x ∥^{2} .

Hence

∥ (I + T) x ∥ \geq ∥ x ∥

. In particular

I + T

is injective and bounded below by one.

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