Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 225 / 3 / b / i / Solution

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