 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/ib/paper-3/10f/i/solution

Codex  0 Created 2026-09-24 Updated 2026-10-06

The real and imaginary parts of the operator are uniquely

α_{1} = \frac{α + α ^{*}}{2}, α_{2} = \frac{α - α ^{*}}{2 i} .

α = α_{1} + i α_{2}

. Conversely such a decomposition has

α^{*} = α_{1} - i α_{2}

. Expanding both products gives

α α^{*} - α^{*} α = - 2 i (α_{1} α_{2} - α_{2} α_{1}) .

Consequently $α$ is a normal operator exactly when its self-adjoint real and imaginary parts commute.

 New to topics? Read the docs here!