On operator factorization of linear relations
type: Item
instance of: scholarly article
title: On operator factorization of linear relations
zbMATH Open document ID: 1298.47005
author: Adrian Sandovici
author: Zoltán Sebestyén
published in: Positivity
publication date: 2014-01-29T00:00:00Z
review text: Let \(\mathcal X\)\(,\mathcal Y\) both be real or complex vector spaces. A linear relation \(\mathcal A\) from \(\mathcal X\) to \(\mathcal Y\) is a subspace of the Cartesian product \(\mathcal X\times \mathcal Y\) (whose elements are denoted by \((x,y)\), \(x \in X\), \(y \in Y\)). Let \(\mathcal L(\mathcal X,\mathcal Y)\) denote the set of all linear relations from \(\mathcal X\) to \(\mathcal Y\) and \(\mathcal O(\mathcal X,\mathcal Y)\) denote the set of all linear transformations (operators) from \(\mathcal X\) into \(\mathcal Y\). For a linear relation \(\mathcal A\) from \(\mathcal X\) to \(\mathcal Y\), define the domain, range, kernel and the multi-valued part of \(\mathcal A\), respectively, by: \(\operatorname{dom}A=\{x: (x,y) \in \mathcal A\}\), \(\operatorname{ran}A=\{y: (x,y) \in \mathcal A\}\), \(\operatorname{ker}A=\{x: (x,0) \in \mathcal A\}\) and \(\operatorname{mul}A=\{y: (0,y) \in \mathcal A\}\), all of which are subspaces of \(\mathcal X\times \mathcal Y\). The main result of the article under review (generalizing a well known result of \textit{R. G. Douglas} [Proc. Am. Math. Soc. 17, 413--415 (1966; Zbl 0146.12503)]) is the following Theorem. Let \(\mathcal X\), \(\mathcal Y\), \(\mathcal Z\) be vector spaces, \(A\in \mathcal L(\mathcal X,\mathcal Z)\) and \(B \in \mathcal L(\mathcal X,\mathcal Y)\). Then there exists \(T \in \mathcal O(\mathcal X,\mathcal Y)\) satisfying \(A=TB\) if and only if \(A,B\) and \(T\) satisfy: \(\operatorname{dom}A \subseteq\operatorname{dom}B\), \(\operatorname{ker}B \subseteq\operatorname{ker}A\), and there exists a subspace \(\mathcal D\subseteq\operatorname{mul}B\) and a surjective linear transformation \(T_{\operatorname{mul}}: \mathcal D\rightarrow\operatorname{mul}A\).
reviewed by: K. C. Sivakumar
Mathematics Subject Classification ID: 47A06
Mathematics Subject Classification ID: 47A05
zbMATH DE Number: 6251995
zbMATH Keywords: linear space
zbMATH Keywords: linear operator
zbMATH Keywords: linear relation
zbMATH Keywords: Douglas factorization
MaRDI profile type: Publication
full work available at URL: s11117-013-0224-x
OpenAlex ID: W1985852804
cites work: Operational calculus of linear relations
cites work: Q3838080
cites work: On Majorization, Factorization, and Range Inclusion of Operators on Hilbert Space
cites work: Factorizations of linear relations
cites work: An index formula for the product of linear relations
cites work: Q3344729
DOI: 10.1007/S11117-013-0224-X
description: scientific article; zbMATH DE number 6251995