On \(\ell_ p\)-summability of the characteristic values of integral operators on \(L_ 2({\mathbb{R}}^ N)\).
type: Item
instance of: scholarly article
title: On \(\ell_ p\)-summability of the characteristic values of integral operators on \(L_ 2({\mathbb{R}}^ N)\).
zbMATH Open document ID: 0637.47014
DOI: 10.1007/BF01196122
published in: Integral Equations and Operator Theory
publication date: 1987-01-01T00:00:00Z
review text: Let \(B_ p(L_ 2)\) \((0<p<\infty)\) denote the space of all compact linear operators K on \(L_ 2({\mathbb{R}}^ N\)) with the following property: the sum of p-th powers of all positive eigenvalues (repeated accordng to their multiplicity) of the nonnegative compact operator (K \(*K)^{1/2}\) converges. We prove a \(B_ p\)-criterion for an integral operator K on \(L_ 2({\mathbb{R}}^ N\)), for \(0<p\leq 2\). We require that its kernel have sufficient fractional smoothness and decay at infinity. Our proof is based on a suitable factorization of the operator K which involves fractional powers \((H+\lambda I)^{-\omega}\), \(\lambda\geq 0\), \(0<\omega <1\), of the resolvent of the selfadjoint elliptic operator \(H=(- \Delta)^ m+M_ q^{2m}\) on \(L_ 2({\mathbb{R}}^ N\)).
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 47B10
Mathematics Subject Classification ID: 47F05
zbMATH DE Number: 4038570
zbMATH Keywords: \(\ell _ p\)-summability of characteristic values
zbMATH Keywords: \(B_ p\)-criterion for an integral operator
zbMATH Keywords: fractional smoothness and decay at infinity
zbMATH Keywords: factorization
zbMATH Keywords: fractional powers
zbMATH Keywords: resolvent of the selfadjoint elliptic operator
author: Peter Takáč
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4038570