 Solution

The two rank-one operators have orthogonal ranges. Their Hilbert-Schmidt inner product is zero, while $\Vert C_i{\Vert }_{\mathrm{HS}}={\lambda }_i$, so the Hilbert-Schmidt distance between covariance operators isFor 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 operatorsFinally $C_2^{1/2}{C}_1^{1/2}=0$, so every singular value in the Procrustes cross-term vanishes. Hence