An extension of Mercer's theorem via Pontryagin spaces
type: Item
instance of: scholarly article
title: An extension of Mercer's theorem via Pontryagin spaces
zbMATH Open document ID: 1290.47047
published in: Integral Equations and Operator Theory
publication date: 2013-04-15T00:00:00Z
review text: Results connected with Mercer's theorem describe series representations for the generating kernel of an integral operator. E.g., if \(T\) is a nonnegative compact operator on \(L^2([a,b])\) with continuous kernel \(K\), then \(K\) can be written in terms of an orthonormal basis of the eigenfunctions \(\varphi_n\) and the eigenvalues \(\lambda_n\) of \(T\), \[ K(x,y) = \sum_{n=1}^{\infty} \lambda_n \varphi_n(x) \overline{ \varphi_n(y)}, \quad x,y \in [a,b]. \] The present paper generalizes this idea. Here, the interval \([a,b]\) is replaced by a locally finite measure space \((X,\nu)\) and \(T\) is replaced by the composition of two self-adjoint integral operators \(R\) and \(S\) on \(L^2(X)\) generated by continuous kernels such that \(R\) is compact and positive and \(S\) is continuous and satisfies some spectral decomposition. The proof of this result is based on a Pontryagin space structure on \(L^2(X)\). The negative subspace of this Pontryagin space is given as the spectral subspace of \(S\) corresponding to the (finitely many) negative eigenvalues of \(S\).
reviewed by: Carsten Trunk
Mathematics Subject Classification ID: 47G10
Mathematics Subject Classification ID: 45H05
Mathematics Subject Classification ID: 47B50
Mathematics Subject Classification ID: 47B38
zbMATH DE Number: 6153937
zbMATH Keywords: Mercer's theorem
zbMATH Keywords: integral operators
zbMATH Keywords: positive operators
zbMATH Keywords: spectral theorem
zbMATH Keywords: Kreǐn spaces
zbMATH Keywords: Pontryagin spaces
author: Valdir A. Menegatto
author: Claudemir P. Oliveira
MaRDI profile type: Publication
full work available at URL: s00020-012-2008-2
OpenAlex ID: W2034103335
cites work: Q3866685
cites work: Q4692935
cites work: Q4702980
cites work: Q4316731
cites work: Eigenvalues of integral operators defined by smooth positive definite kernels
cites work: On the nuclearity of integral operators
cites work: Q3916999
cites work: Q4761216
DOI: 10.1007/S00020-012-2008-2
description: scientific article; zbMATH DE number 6153937