Let T be a self-adjoint operator on a finite-dimensional Hilbert space. It is shown that the distribution of the eigenvalues of a compression of T to a subspace of a given dimension is almost the same for almost all subspaces. This is a coordinate-free analogue of a recent result of Chatterjee and Ledoux on principal submatrices. The proof is based on measure concentration and entropy techniques, and the result improves on the result of Chatterjee and Ledoux in various ways. This is joint work with Mark Meckes.