On properties of the \(\xi \)-function in semi-finite von Neumann algebras
type: Item
instance of: scholarly article
title: On properties of the \(\xi \)-function in semi-finite von Neumann algebras
zbMATH Open document ID: 1197.47026
author: Anna Skripka
published in: Integral Equations and Operator Theory
publication date: 2009-10-19T00:00:00Z
review text: Let \(\mathcal A\) be a semi-finite von Neumann algebra and \(\tau\) a normal faithful semifinite trace on it. One of the main objectives of the present paper is to prove versions of the Birman-Schwinger principle for (relative) trace class perturbation problems of dissipative operators in \(\mathcal A\) and selfadjoint operators affiliated with \(\mathcal A\). E.g., it is shown that for bounded and boundedly invertible \(M=M^*\) in \(\mathcal A\), a selfadjoint operator \(H_0\) affiliated with \(\mathcal A\) and \(K\) in the \(\tau\)-trace ideal \(\mathcal L_2(\mathcal A,\tau)\), the relation \[ \xi\bigl(\lambda,H_0,H_0+KM^{-1}K^*\bigr)=\lim_{\varepsilon\rightarrow 0+} \xi\bigl(M+K^*(H_0-\lambda-i\varepsilon)^{-1}K,M\bigr) \] holds for a.e.\ \(\lambda\in\mathbb R\). This principle is then further generalized to the case when \(\lambda\) is in the spectrum of the perturbed or unperturbed operator. Moreover, if the normal boundary values \(K^*(H_0-\lambda-i0)^{-1}K\) exist in the operator norm with imaginary parts in the \(\tau\)-trace class, the function \(\xi(\lambda,H_0,H_1)\) is related to the scattering operator associated with the pair \(\{H_0-\lambda,H-\lambda\}\).
reviewed by: Jussi Behrndt
Mathematics Subject Classification ID: 47A55
Mathematics Subject Classification ID: 47C15
Mathematics Subject Classification ID: 46L52
zbMATH DE Number: 5617569
zbMATH Keywords: spectral shift function
zbMATH Keywords: dissipative operator
zbMATH Keywords: semi-finite von Neumann algebra
MaRDI profile type: Publication
full work available at URL: s00020-008-1617-2
OpenAlex ID: W2082047398
DOI: 10.1007/S00020-008-1617-2
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description: scientific article; zbMATH DE number 5617569