What is Rotation-equivariant estimator, nonlinear shrinkage?

A rotation-equivariant estimator keeps the sample eigenvectors and changes only the eigenvalues: $\hat\Sigma = U\operatorname{diag}(d)U^\top$. Nonlinear shrinkage chooses each $d_i$ separately, as an estimate of the oracle value $u_i^\top\Sigma u_i$, the true variance of the $i$-th sample eigenportfolio.