The two rank-one operators have orthogonal ranges. Their Hilbert-Schmidt inner product is zero, while $\lVert C_i\rVert_{\mathrm{HS}}=\lambda_i$, so the <Hilbert-Schmidt distance between covariance operators> is
$$
d_L(C_1,C_2)=\sqrt{\lambda_1^2+\lambda_2^2}.
$$
For each $C_i$, the <positive square root of an operator> is $C_i^{1/2}=\sqrt{\lambda_i}\,e_i\otimes e_i$. Orthogonality therefore gives the <square-root distance between covariance operators>
$$
d_R(C_1,C_2)=\sqrt{\lambda_1+\lambda_2}.
$$
Finally $C_2^{1/2}C_1^{1/2}=0$, so every singular value in the Procrustes cross-term vanishes. Hence
$$
d_P(C_1,C_2)=\sqrt{\lambda_1+\lambda_2}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 225 / 3 / c / Solution

{title2=$d_L$}

The linear distance between covariance operators is $d_L(C_1,C_2)=\lVert C_1-C_2\rVert_{\mathrm{HS}}$.


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