 Solution

ID: past-exam-of-the-analysis-course-of-the-university-of-oxford/2011/3/a/iv/solution

T^{n} - λ^{n} I = (T - λ I) (T^{n - 1} + λ T^{n - 2} + \dots + λ^{n - 1} I)

has commuting factors because both are polynomials in

T

. If

λ^{n} \in / σ (T^{n})

, their product is invertible. The preceding part would then make

T - λ I

invertible, contradicting

λ \in σ (T)

. Therefore

λ \in σ (T) ⟹ λ^{n} \in σ (T^{n}) .

This proves the required inclusion without assuming the full spectral mapping theorem.

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