Positive integral operators in unbounded domains
type: Item
instance of: scholarly article
title: Positive integral operators in unbounded domains
zbMATH Open document ID: 1068.47060
author: Jorge Buescu
published in: Journal of Mathematical Analysis and Applications
publication date: 2004-08-20T00:00:00Z
review text: The author considers a generalization to the case of the entire real line (\(I={\mathbb R}\)) of well-known results on the representation of positive integral operators with continuous kernel \(k(x,y)\) acting in \(L^2(I)\), \({\mathcal K} \phi=\int_I k(x,y) \phi(y) dy\), where \(I\) is a compact subinterval of \({\mathbb R}\). The main results are as follows: If \({\mathcal K}\) is positive and \(k(x,x)\in L^1({\mathbb R})\), then \({\mathcal K}\) is Hilbert-Schmidt and therefore compact. If, in addition, \(k(x,x)\rightarrow 0\) as \(| x| \rightarrow \infty\), then \(k\) is represented by an absolutely and uniformly convergent bilinear series of continuous eigenfunctions and \({\mathcal K}\) is trace class (Theorem 1). Replacing the first assumption by the stronger one \(k^{1/2} (x,x)\in L^1({\mathbb R})\), one gets that then \(k\in L^1({\mathbb R})\) and the bilinear series also converges in \(L^1({\mathbb R})\) (Theorem 2). Sharp norm bounds are given and Mercer's theorem is derived as a special case.
Mathematics Subject Classification ID: 47G10
Mathematics Subject Classification ID: 45P05
Mathematics Subject Classification ID: 47B10
zbMATH DE Number: 2093829
zbMATH Keywords: positive integral operator with continuous kernel
zbMATH Keywords: Hilbert-Schmidt operator
zbMATH Keywords: compactness
zbMATH Keywords: norm bounds
zbMATH Keywords: Mercer's theorem
reviewed by: A. B. Lebre
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.04.007
OpenAlex ID: W2054972504
cites work: Theory of Reproducing Kernels
cites work: Representation of kernels of integral operators by bilinear series
DOI: 10.1016/J.JMAA.2004.04.007
description: scientific article; zbMATH DE number 2093829