Another observation about operator compressions

TL;DR

This paper demonstrates that the eigenvalue distribution of a compressed self-adjoint operator is nearly identical across most subspaces, extending previous principal submatrix results using measure concentration and entropy methods.

Contribution

It provides a coordinate-free, almost-sure eigenvalue distribution similarity result for operator compressions, improving upon prior principal submatrix findings.

Findings

Eigenvalue distributions are nearly identical for most subspaces.

The result applies to self-adjoint operators on finite-dimensional spaces.

The proof employs measure concentration and entropy techniques.

Abstract

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 some aspects of the result of Chatterjee and Ledoux.