Another observation about operator compressions

@article{Meckes2010AnotherOA,
  title={Another observation about operator compressions},
  author={Elizabeth S. Meckes and Mark W. Meckes},
  journal={arXiv: Probability},
  year={2010}
}
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. 
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References

SHOWING 1-10 OF 15 REFERENCES
An observation about submatrices
Let $M$ be an arbitrary Hermitian matrix of order $n$, and $k$ be a positive integer less than $n$. We show that if $k$ is large, the distribution of eigenvalues on the real line is almost the same
The Sizes of Compact Subsets of Hilbert Space and Continuity of Gaussian Processes
The first two sections of this paper are introductory and correspond to the two halves of the title. As is well known, there is no complete analog of Lebesue or Haar measure in an
Majorizing measures: the generic chaining
Majorizing measures provide bounds for the supremum of stochastic processes. They represent the most general possible form of the chaining argument going back to Kolmogorov. Majorizing measures arose
Approximation of Projections of Random Vectors
Let X be a d-dimensional random vector and Xθ its projection onto the span of a set of orthonormal vectors {θ1,…,θk}. Conditions on the distribution of X are given such that if θ is chosen according
Weak Convergence and Empirical Processes: With Applications to Statistics
TLDR
This chapter discusses Convergence: Weak, Almost Uniform, and in Probability, which focuses on the part of Convergence of the Donsker Property which is concerned with Uniformity and Metrization.
Topics in Optimal Transportation
Introduction The Kantorovich duality Geometry of optimal transportation Brenier's polar factorization theorem The Monge-Ampere equation Displacement interpolation and displacement convexity Geometric
The concentration of measure phenomenon
Concentration functions and inequalities Isoperimetric and functional examples Concentration and geometry Concentration in product spaces Entropy and concentration Transportation cost inequalities
Matrix Analysis
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