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The Procrustes distance between covariance operators isFor unitary $R$,The polar decomposition of a bounded operator impliesTherefore

For positive trace-class operators, the square-root distance between covariance operators isIf $C_i=L_i{L}_i^{\ast }$ are Hilbert-Schmidt factorizations, the Procrustes distance between covariance operators iswhere $\mathcal{O}(H)$ is the group of orthogonal operators on the real separable Hilbert space $H$.

For positive trace-class covariance operators $C_i=L_i{L}_i^{\ast }$, their Procrustes distance between covariance operators iswhere $\mathcal{O}(H)$ is the group of unitary operators on the real Hilbert space $H=L^2[0,1]$. Expanding the square gives the equivalent formula

For factorizations $C_i=L_i{L}_i^{\ast }$ and $C_j=L_j{L}_j^{\ast }$, the Procrustes distance between covariance operators is the following infimum over unitary operators $R$:Unitary invariance of the Hilbert-Schmidt norm givesThe polar decomposition of a bounded operator and trace duality implywhere the last equality expresses the trace norm as the sum of the singular values. Taking the infimum proves