s-numbers of singular integrals for the invariance of absolutely continuous spectra in fractional dimensions
type: Item
instance of: scholarly article
title: s-numbers of singular integrals for the invariance of absolutely continuous spectra in fractional dimensions
zbMATH Open document ID: 0732.47023
DOI: 10.1016/0022-1236(90)90026-H
author: Guy David
author: Dan Voiculescu
published in: Journal of Functional Analysis
publication date: 1990-01-01T00:00:00Z
review text: Let K be a \(C^{\infty}\)-function on \({\mathbb{R}}^ n\setminus \{0\}\) such that \(K(\lambda x)=\lambda^{-\alpha}K(x)\) for some \(\alpha >0\). Let \(\mu\) be a compactly supported positive Radon measure on \({\mathbb{R}}^ n\), and define \[ h(r):=\sup \{\mu (B(x,r)):\;x\in {\mathbb{R}}^ n\}, \] where B(x,r) is the open ball with center x and radius \(r>0\). Then \[ T: f(x)\to \int K(x-y)f(y)d\mu (y) \] defines a compact operator on \(L^ 2({\mathbb{R}}^ n,\mu)\). The authors show how the asymptotic behaviour of the s-numbers \(s_ n(T)\) depends on \(\alpha\) and on the behaviour of h(r) as \(r\to 0\). An application to the kernels \(K_ j(x)=x_ j/| x|^ 2\) yields results about the invariance of the absolutely continuous spectrum of the n-tuple of multiplication operators \(f(x)\to x_ jf(x)\) with respect to perturbation by operators belonging to certain normed ideals. The paper is readable for insiders only.
Mathematics Subject Classification ID: 47B06
Mathematics Subject Classification ID: 47B38
zbMATH DE Number: 4210878
zbMATH Keywords: compactly supported positive Radon measure
zbMATH Keywords: asymptotic behaviour of the s-numbers
zbMATH Keywords: kernels
zbMATH Keywords: absolutely continuous spectrum
zbMATH Keywords: multiplication operators
reviewed by: Albrecht Pietsch
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4210878