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Exercise 6.5.9

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The operator $U$ may not be unitary. For example, let $U (a, b) = (a + b, 0)$ be an operator on $ℂ^{2}$ . Pick the basis ${(1, 0), (0, 1)}$ and we may observe that

∥ U (1, 0) ∥ = ∥ U (0, 1) ∥ = 1 = ∥ (1, 0) ∥ = ∥ (0, 1) ∥ .

But it is not unitary since

∥ U (1, 1) ∥ = 1 \neq ∥ (1, 1) ∥ = \sqrt{2} .

jephianlin

2011-06-27 00:00

Exercise 6.5.9

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