Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 3 / 10F / i

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- Solution i

Solution

The real and imaginary parts of the operator are uniquely

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

They are self-adjoint operators, and

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.

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