= Solution
For factorizations $C_i=L_iL_i^*$ and $C_j=L_jL_j^*$, the <Procrustes distance between covariance operators> is the following infimum over unitary operators $R$:
$$
d_P(C_i,C_j)=\inf_R\lVert L_i-L_jR\rVert_{\mathrm{HS}}.
$$
Unitary invariance of the Hilbert-Schmidt norm gives
$$
\lVert L_i-L_jR\rVert_{\mathrm{HS}}^2
=\lVert L_i\rVert_{\mathrm{HS}}^2+\lVert L_j\rVert_{\mathrm{HS}}^2
-2\operatorname{Re}\operatorname{tr}(L_i^*L_jR).
$$
The <polar decomposition of a bounded operator> and trace duality imply
$$
\sup_R\operatorname{Re}\operatorname{tr}(L_i^*L_jR)
=\lVert L_j^*L_i\rVert_1
=\sum_{k=1}^{\infty}\sigma_k,
$$
where the last equality expresses the <trace norm> as the sum of the <singular values>. Taking the infimum proves
$$
d_P(C_i,C_j)^2
=\lVert L_i\rVert_{\mathrm{HS}}^2+\lVert L_j\rVert_{\mathrm{HS}}^2-2\sum_{k=1}^{\infty}\sigma_k.
$$
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