Let $X$ be a random vector taking values in a separable Hilbert space $H$ such that $E\|X\|^2<\infty$ and $E X=\mu$. Then the covariance operator $R\colon H\to H$ is defined by the formula \begin{equation*} Rx:=E\langle x,X-\mu\rangle (X-\mu)=E\overline{\langle X-\mu,x\rangle}(X-\mu) \end{equation*} for $x\in H$, so that for any vectors $x$ and $y$ in $H$, \begin{equation*} \langle Rx,y\rangle=E\overline{\langle X-\mu,x\rangle}\langle X-\mu,y\rangle, \end{equation*} the covariance of the "$x$-coordinate" $\langle X,x\rangle$ and the "$y$-coordinate" $\langle X,y\rangle$ of the random vector $X$.
A brief account on the compactness of the covariance operator and its spectral decomposition is given in Appendices E and F of the paper at \url{http://projecteuclid.org/euclid.ejs/1460463653}.