 Covariance operator of a Gaussian measure

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