On the removal of finite discrete spectrum by coefficient stripping
type: Item
instance of: scholarly article
title: On the removal of finite discrete spectrum by coefficient stripping
zbMATH Open document ID: 1277.47046
DOI: 10.4171/JST/3
author: B. Simon
published in: Journal of Spectral Theory
publication date: 2013-01-23T00:00:00Z
review text: The main goal of this paper is to prove the following result. Let \(J\) be a bounded self-adjoint operator on a Hilbert space \(\mathcal{H}\), and let \((P_{n}), (Q_{n}), (R_{n})\) be sequences of orthogonal projections in \(\mathcal{H}\) satisfying \[ P_{n} Q_{n} = Q_{n} P_{n} = P_{n} R_{n} = R_{n} P_{n} = Q_{n} R_{n} = R_{n} Q_{n} = 0, \] \[ P_{n} + Q_{n} + R_{n} = I, \] \[ s - \lim_{n \to \infty} Q_{n} = s - \lim_{n \to \infty} P_{n} = 0, \] \[ P_{n} J Q_{n} = Q_{n} J P_{n} = 0. \] It is stated that, if the spectrum of \(J\) on the interval \((- \infty, 0)\) consists only of finitely many eigenvalues of finite multiplicity, then, for sufficiently large \(n\), \(Q _{n} J Q_{n}\) is a non-negative operator. The result is applied to block Jacobi matrices and also to Sturm-Liouville operators with operator-valued coefficients.
reviewed by: Petru A. Cojuhari
Mathematics Subject Classification ID: 47B36
Mathematics Subject Classification ID: 34L15
Mathematics Subject Classification ID: 81Q10
Mathematics Subject Classification ID: 47A10
zbMATH DE Number: 6128401
zbMATH Keywords: coefficient stripping
zbMATH Keywords: block Jacobi matrices
MaRDI profile type: Publication
cites work: Discrete Sturm-Liouville operators and some problems of function theory
cites work: Q5387010
cites work: Q5718741
full work available at URL: https://doi.org/10.4171/jst/3
OpenAlex ID: W2076228937
description: scientific article; zbMATH DE number 6128401