Covariance operator of a Gaussian measure
= Covariance operator of a Gaussian measure
The covariance operator $K$ of a Gaussian measure on a real Hilbert space satisfies $\langle Kh,g\rangle=\operatorname{Cov}(\langle U,h\rangle,\langle U,g\rangle)$. A positive self-adjoint operator is the covariance of a Hilbert-space-valued Gaussian random variable exactly when it is trace class.