Eigenvalue decay rates for positive integral operators
type: Item
instance of: scholarly article
title: Eigenvalue decay rates for positive integral operators
zbMATH Open document ID: 1296.45001
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 2013-12-06T00:00:00Z
review text: Positive definite integral operators in \(L^2(\Omega,\mathbb C)\) with a metric space \(\Omega\) are considered. In the first part it is shown that if the kernel is continuous with a square-integrable diagonal then it has an series representation in terms of eigenfunctions with a convergent series of singular values (Mercer's theorem); under the hypothesis that a series representation exists, the latter is shown also without the hypothesis of positive definiteness. In the second part, similar results for the Schatten class are established where, roughly speaking, the continuity hypothesis is replaced by two types of hypothesis: It is required that large balls in \(\Omega\) can be decomposed into (at most) a certain number of pieces satisfying diameter and measure estimates (with appropriate powers), and after a decomposition of \(\Omega\) into a finite number of pieces, the values \(k(x,y)\) of the kernel on the side-diagonal depend in a certain sense Hölder continuous from the average of the diagonal values \(k(x,x)\) and \(k(y,y)\). Actually, the results about the Schatten \(p\)-class are only a corollary, since a more precise decay estimate for the eigenvalues is established.
Mathematics Subject Classification ID: 45C05
Mathematics Subject Classification ID: 47B10
Mathematics Subject Classification ID: 45P05
Mathematics Subject Classification ID: 47G10
Mathematics Subject Classification ID: 47A70
zbMATH DE Number: 6236079
zbMATH Keywords: eigenvalue decay rates
zbMATH Keywords: positive definiteness
zbMATH Keywords: Mercer's theorem
zbMATH Keywords: Schatten class
zbMATH Keywords: Hilbert-Schmidt operators
zbMATH Keywords: positive definite integral operator
zbMATH Keywords: series representation in terms of eigenfunctions
zbMATH Keywords: series of singular values
author: José C. Ferreira
author: Valdir A. Menegatto
MaRDI profile type: Publication
full work available at URL: s10231-012-0256-z
OpenAlex ID: W2092798400
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DOI: 10.1007/S10231-012-0256-Z
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description: scientific article; zbMATH DE number 6236079