The <Procrustes distance between covariance operators> is
$$
d_P(C_1,C_2)=\inf_{R\ {\rm unitary}}
\lVert L_1-L_2R\rVert_{\rm HS}.
$$
For unitary $R$,
$$
\lVert L_1-L_2R\rVert_{\rm HS}^2
=\lVert L_1\rVert_{\rm HS}^2+\lVert L_2\rVert_{\rm HS}^2
-2\operatorname{Re}\operatorname{tr}(R^*L_2^*L_1).
$$
The <polar decomposition of a bounded operator> implies
$$
\sup_R\operatorname{Re}\operatorname{tr}(R^*L_2^*L_1)
=\lVert L_2^*L_1\rVert_1=\sum_{k=1}^\infty\sigma_k.
$$
Therefore
$$
\boxed{d_P(C_1,C_2)^2
=\lVert L_1\rVert_{\rm HS}^2+\lVert L_2\rVert_{\rm HS}^2
-2\sum_{k=1}^\infty\sigma_k}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 225 / 3 / iii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-225/3/iii/solution