On generalized resolvents of Hermitian relations in Krein spaces
type: Item
instance of: scholarly article
title: On generalized resolvents of Hermitian relations in Krein spaces
zbMATH Open document ID: 0955.47022
DOI: 10.1007/BF02366102
author: V. A. Derkach
published in: Journal of Mathematical Sciences (New York)
publication date: 2000-03-28T00:00:00Z
review text: Hermitian and selfadjoint relations and their generalized resolvents have been studied in various papers [see e. g. \textit{V. Derkach}, Integral Equations Oper. Theory 23, No. 4, 387-415 (1995; Zbl 0837.47030) and \textit{A. Dijksma, H. Langer} and \textit{H. de Snoo}, Lect. Notes Math. 1242, 1-42 (1987; Zbl 0628.47026); Oper. Theory, Adv. Appl. 48, 261-274 (1990; Zbl 0734.47018)]. This paper is an important continuation of that work. Let \(A\) be a closed Hermitian linear relation on a Krein space \(\mathcal K\), i. e. \(A\subset A^+\), where \(A^+\) is the Krein space adjoint of \(A\). A triple \(\Pi=\{{\mathcal H}, \Gamma _1,\Gamma _2\}\) is called a boundary-value space (BVS) of \(A^+\) if \(\mathcal H\) is a Hilbert space, \(\Gamma =(\Gamma _1,\Gamma_2)\) is a bounded operator from \(A^+\) onto \({\mathcal H}\oplus{\mathcal H}\), and \[ [f',g]-[f,g']=(\Gamma _1 \widehat f,\Gamma _2 \widehat g)_{\mathcal H} -(\Gamma _2 \widehat f,\Gamma _1 \widehat g)_{\mathcal H} \] for all \(\widehat f=\{f,f'\}.\widehat g=\{g,g'\}\in A^+\). There is a one-to-one relation between extensions \(\widetilde A\) of \(A\), \(A\subset \widetilde A\subset A^+\), and subspaces \(\theta\) of \({\mathcal H}\oplus{\mathcal H}\) given by \(\theta=\Gamma \widetilde A\), \(\widetilde A=\widetilde A_\theta=\Gamma ^{-1}\theta\). Each BVS has two natural selfadjoint extensions \(A_i=\text{ker}\Gamma _i\) (\(i=1,2\)). Let \({\mathcal N}_\lambda={\mathcal K}[-]{\mathcal R}(A-\overline{\lambda })\), \(\widehat{\mathcal N}_\lambda=\{\{f_\lambda ,\lambda f_\lambda \}\mid f_\lambda \in {\mathcal N}_\lambda\}\). The operator-valued function \(M(\lambda)\) defined on \(\rho(A_2)\) by \(M(\lambda)\Gamma _2\widehat f_\lambda =\Gamma _1 f_\lambda \), \(\lambda \in \rho(A_2)\), \(\widehat f_\lambda \in \widehat{\mathcal N}_\lambda \), is called the Weyl function of \(A\) corresponding to the BVS \(\Pi=\{{\mathcal H}, \Gamma _1,\Gamma _2\}\). The Weyl function contains all information about the spectrum of …
Mathematics Subject Classification ID: 47B50
Mathematics Subject Classification ID: 46C20
Mathematics Subject Classification ID: 47A06
Mathematics Subject Classification ID: 47A56
Mathematics Subject Classification ID: 47A57
zbMATH DE Number: 1423195
zbMATH Keywords: boundary-value space
zbMATH Keywords: Weyl function
zbMATH Keywords: generalized resolvent
zbMATH Keywords: Hermitian linear relation
zbMATH Keywords: Krein space
zbMATH Keywords: operator-valued function
zbMATH Keywords: definitizable
zbMATH Keywords: truncated moment problem
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1423195