Asymptotic expansions for spectral convergence of compact self-adjoint operators on general spectral subsets, with application to kernel Gram matrices

TL;DR

This paper develops asymptotic expansions for the spectral convergence of compact self-adjoint operators, with applications to kernel Gram matrices, providing new concentration inequalities and weak convergence results under easily verifiable assumptions.

Contribution

It introduces general asymptotic expansions for eigenvalues and eigenprojections of operators on arbitrary spectral subsets, extending existing results and simplifying assumptions.

Findings

Derived asymptotic expansions for eigenvalues and eigenprojections.

Established concentration inequalities for kernel Gram matrices.

Proved weak convergence results under minimal kernel assumptions.

Abstract

We study the spectral convergence of compact, self-adjoint operators on a separable Hilbert space under operator norm perturbations, and derive asymptotic expansions for their eigenvalues and eigenprojections. Our analysis focuses on eigenvalues indexed by a general subset, with minimal restrictions on their selection. The usefulness of the provided expansions is illustrated by an application to kernel Gram matrices, deriving concentration inequalities as well as weak convergence results, which, in contrast to existing literature, are primarily relying on assumptions on the kernel that are easy to check.