Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 225 / 1 / b / Solution

For every orthonormal basis $(e_j)$ of the infinite-dimensional Hilbert space $L^2[0,1]$, the identity operator satisfiesPart a shows that every covariance operator of a square-integrable Hilbert-space random element is trace-class. Therefore the identity cannot be a covariance operator.