Empirical covariance operator ($\hat{C}$)

For observations $X_1,\dots ,X_n$ and a chosen center $m$, the empirical covariance operator is ${\hat{C}}_m=n^{-1}{\sum }_i(X_i-m)\otimes (X_i-m)$. Its expectation equals the population covariance plus the rank-one operator formed from the centering error.