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

Eunseong Bae; Wolfgang Polonik

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arXiv · 2602.00999

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

Eunseong Bae·Wolfgang Polonik

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Abstract

We study the spectral convergence of compact, self-adjoint operators on a separable Hilbert space, and derive the first-order asymptotic expansions for their eigenvalues, eigenvectors and eigenprojections, along with remainder bounds expressed in terms of weighted perturbation quantities. Our analysis focuses on eigenvalues indexed by a general subset, with minimal restrictions on their selection. In particular, when the eigenvalues of interest are clustered, i.e., the inner spectral gaps are small, we provide different types of expansions for the eigenvalues, with remainder bounds that are robust to the inner spectral gaps; this contrasts with the existing literature, which mainly focuses on isolated eigenvalues. 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, primarily rely on assumptions on the kernel that are easy to check.

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Eunseong Bae, Wolfgang Polonik. 2026-02-01. First-order asymptotic expansions for spectral convergence of compact self-adjoint operators on general spectral subsets, with application to kernel Gram matrices. https://arxiv.org/abs/2602.00999

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