The limit spectrum of a positive operator in \(L_{2}\) that is integral on some subspace
type: Item
instance of: scholarly article
title: The limit spectrum of a positive operator in \(L_{2}\) that is integral on some subspace
zbMATH Open document ID: 1200.47043
author: V. B. Korotkov
published in: Siberian Mathematical Journal
publication date: 2010-11-15T00:00:00Z
review text: Let \((X,\Omega,\mu)\) be a finite positive measure space which is non-atomic and let \(L^2(X)\) be a separable Hilbert space with norm \(\|\cdot\|_2\) and inner product \((\cdot|\cdot)\). If \(\Delta\) is a subspace of \(L^2(X)\), then an operator \({\mathcal K}:\Delta\to L^2(X)\) is defined to be an integral operator if there is a kernel \(K\) on \(X\times X\) for which the operator \({\mathcal K}\) may be represented in the form \[ {\mathcal K}(f)(x)= \int_X K(x,y) f(y)\,d\mu(y). \] A linear operator \(M: \text{Dom}(M)\to L^2(X)\) with dense domain \(\text{Dom}(M)\) has a limit spectrum which contains \(0\) (that is, \(0\in\sigma_e(M)\)) if there is an orthonormal sequence \(\{h_j\), \(j= 0,1,2,\dots\}\) such that \(\lim_{j\to\infty}\| M(h_j)\|_2= 0\). The main results of this paper include statements that, if \(S: L^2(X)\to L^2(X)\) is representable as an integral operator on a dense subspace \(Do(S)\) of \(L^2(X)\) and \(Do(S)\) is such that \(\chi_E Do(S)\subseteq Do(S)\) for each characteristic function \(\chi_E\) of a measurable set \(E\), then \(0\in\sigma_e(S)\). Examples provided in the introduction of the paper indicate that representations as integral operators alone would not imply \(0\in\sigma_e(S)\).
reviewed by: George O. Okikiolu
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 47G10
Mathematics Subject Classification ID: 47A10
zbMATH DE Number: 5816259
zbMATH Keywords: limit spectrum
zbMATH Keywords: positive operator
zbMATH Keywords: integral operator
MaRDI profile type: Publication
full work available at URL: s11202-010-0046-z
OpenAlex ID: W2072209111
cites work: Q4334625
cites work: Q4759674
cites work: Q4057194
cites work: Q3041809
cites work: Q3669960
cites work: Application of methods of the theory of order-bounded operators to the theory of operators inLp-spaces
cites work: Q3218622
cites work: Q3692319
DOI: 10.1007/S11202-010-0046-Z
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description: scientific article; zbMATH DE number 5816259