Solution
= Solution
For positive trace-class operators, the <square-root distance between covariance operators> is
$$
d_R(C_1,C_2)=\lVert C_1^{1/2}-C_2^{1/2}\rVert_{\rm HS}.
$$
If $C_i=L_iL_i^*$ are Hilbert-Schmidt factorizations, the <Procrustes distance between covariance operators> is
$$
d_P(C_1,C_2)
=\inf_{R\in\mathcal O(H)}\lVert L_1-L_2R\rVert_{\rm HS},
$$
where $\mathcal O(H)$ is the group of orthogonal operators on the real separable Hilbert space $H$.