Eigenvalues of integral operators on \(L_ 2(I)\) given by analytic kernels
type: Item
instance of: scholarly article
title: Eigenvalues of integral operators on \(L_ 2(I)\) given by analytic kernels
zbMATH Open document ID: 0790.45002
DOI: 10.1007/BF01225215
author: S. Ganapathi Raman
author: R. Vittal Rao
published in: Integral Equations and Operator Theory
publication date: 1994-04-10T00:00:00Z
review text: The authors consider symmetric Hilbert-Schmidt operators on \(L^ 2(I)\), where \(I\) is an interval in \(\mathbb{R}\). Let \(\{\lambda_ n\}\), \(n\geq 0\), be the corresponding sequence of eigenvalues. It is shown that if \(\ell^ n \lambda_ n \to 0\) as \(n \to \infty\) for \(\ell=1,2,\dots,\) then the operator is unitarily equivalent to an operator with analytic kernel. Conversely, if the operator has an analytic kernel, then \(\ell^{n(\varepsilon)} \lambda_ n \to 0\) as \(n \to \infty\) for \(\ell=1,2,\dots\), where \(n(\varepsilon)=n^{1-\varepsilon}\) and \(\varepsilon>0\). The results are sharp.
Mathematics Subject Classification ID: 45C05
Mathematics Subject Classification ID: 45P05
Mathematics Subject Classification ID: 47A75
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 47G10
zbMATH DE Number: 515558
zbMATH Keywords: symmetric Hilbert-Schmidt operators
zbMATH Keywords: eigenvalues
zbMATH Keywords: analytic kernel
reviewed by: Georgi E. Karadzhov
MaRDI profile type: Publication
cites work: Q3223360
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description: scientific article; zbMATH DE number 515558