Symmetric relations and module spaces
type: Item
instance of: scholarly article
title: Symmetric relations and module spaces
zbMATH Open document ID: 0803.47002
DOI: 10.1007/BF01192457
author: A. V. Shtraus
author: A. Dijksma
author: H. S. V. de Snoo
published in: Integral Equations and Operator Theory
publication date: 1994-08-18T00:00:00Z
review text: \(S\) is said to be a closed linear relation in a Hilbert space \(\mathfrak R\) if \(S\) is a closed linear subspace of the orthogonal direct sum \({\mathfrak R}\oplus {\mathfrak R}\). If \(S\) is a graph of a linear operator then it is identified with this operator. Write: \(\Lambda(S)= \{\ell\in \mathbb{C}\mid (S-\ell)^{-1}\) is a bounded linear operator\}, where \[ (S- \ell)^{- 1}= \bigl\{\{g- \ell f,f\}\in {\mathfrak R}\oplus {\mathfrak R}\mid \{f,g\}\in S\bigr\}. \] Let \(\mathfrak K\) be a non-trivial closed subspace of \(\mathfrak R\). If \[ \Omega({\mathfrak K})= \bigl\{\ell\in \Lambda(S)\mid {\mathfrak R}=\text{ran}(S- \ell)^{-1}+ {\mathfrak K},\text{ direct sum}\bigr\}\neq\emptyset, \] then \(\mathfrak K\) is said to be a module space. Several properties of module spaces are given without the requirement that defect spaces are finite-dimensional. Particular attention is paid to module spaces for symmetric relations.
Mathematics Subject Classification ID: 47A05
zbMATH DE Number: 604800
zbMATH Keywords: closed linear relation
zbMATH Keywords: module spaces for symmetric relations
reviewed by: Danuta Przeworska-Rolewicz
MaRDI profile type: Publication
cites work: Q3973560
cites work: Self-adjoint extensions of symmetric subspaces
cites work: Symmetric and selfadjoint relations in Krein spaces II
cites work: Über einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume \(\Pi_k\). zusammenhängen. II. Verallgemeinerte Resolventen, u-Resolventen und ganze Operatoren
cites work: L-resolvent matrices of symmetric linear relations with equal defect numbers; applications to canonical differential relations
full work available at URL: https://doi.org/10.1007/bf01192457
OpenAlex ID: W2091044783
description: scientific article; zbMATH DE number 604800