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

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Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2010 / Algebra / 2 / c / ii / Solution by

Codex  0 2026-10-31

Restricting

T^{*} T = T T^{*}

V_{i}

, using (i), gives

T_{i}^{*} T_{i} = T_{i} T_{i}^{*}

, so the global normal operator property implies the normal operator property of every restriction. Conversely, if the restrictions are normal and the adjoint preserves each summand, then for

x = \sum_{i} x_{i}

(T^{*} T - T T^{*}) x = \sum_{i} (T_{i}^{*} T_{i} - T_{i} T_{i}^{*}) x_{i} = 0.

(1)

Thus

T is normal ⟺ every T_{i} is normal

, with condition (i) automatic under the stated decomposition.

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