Unitary colligations of operators in Krein spaces
type: Item
instance of: scholarly article
title: Unitary colligations of operators in Krein spaces
zbMATH Open document ID: 0709.47039
DOI: 10.1007/BF01732319
published in: Integral Equations and Operator Theory
publication date: 1990-01-01T00:00:00Z
review text: A colligation of operators in Hilbert spaces is a set \(\delta\) consisting of three Hilbert spaces: \({\mathcal H}_ 1\), \({\mathcal H}_ 2\) and \({\mathcal H}_ 0\), and four linear operators: \(A_{11}\in {\mathcal L}({\mathcal H}_ 0,{\mathcal H}_ 2)\), \(A_{12}\in {\mathcal L}({\mathcal H}_ 1,{\mathcal H}_ 2)\), \(A_{21}\in {\mathcal L}({\mathcal H}_ 0,{\mathcal H}_ 0)\) and \(A_{22}\in {\mathcal L}({\mathcal H}_ 1,{\mathcal H}_ 0)\). A colligation \(\delta\) is said to be unitary if the operator matrix \(A=[A_{ij}]_{i,j=1,2}\) is a unitary operator from \({\mathcal H}_ 0\oplus {\mathcal H}_ 1\) to \({\mathcal H}_ 2\equiv {\mathcal H}_ 0\). When Hilbert spaces \({\mathcal H}_ 1\), \({\mathcal H}_ 2\) and \({\mathcal H}_ 0\) are replaced by Krein spaces \({\mathcal K}_ 1\), \({\mathcal K}_ 2\) and \({\mathcal K}_ 0\) and unitarity is understood in the sense of Krein space structure, we can speak of a unitary colligation of operators in Krein spaces. The characteristic function \(\Theta_{\delta}(z)\) of \(\delta\) is the \({\mathcal L}({\mathcal H}_ 1,{\mathcal H}_ 2)\)-valued analytic function defined by \[ \Theta_{\delta}(z)=A_{12}+zA_{11}(I-zA_{21})^{-1} A_{22}. \] It is known that a \({\mathcal L}({\mathcal H}_ 1,{\mathcal H}_ 2)\)- valued function, defined and analytic on the open unit disk \({\mathbb{D}}\) coincides with the characteristic function of a unitary colligation if and only if \(\Theta\) (z) is contractive, i.e. \(\| \Theta (z)\| \leq 1\) for all z in \({\mathbb{D}}.\) Let \({\mathcal K}_ 1\), \({\mathcal K}_ 2\) be Krein spaces and \(\Theta\) (z) a \({\mathcal L}({\mathcal K}_ 1,{\mathcal K}_ 2)\)-valued bounded function, defined and analytic on \({\mathbb{D}}\). The author constructs a unitary colligation of operators in Krein spaces whose characteristic function coincides with \(\Theta\) (z). Moreover he gives a new proof for a McEnnis' characterization for \(\Theta\) (z) to be written in the form \[ \Theta (z)=\{-TH_ T+zD_{T^*}(I-zT^*)^{-1} D_ T…
reviewed by: T. Ando
Mathematics Subject Classification ID: 47B50
Mathematics Subject Classification ID: 47A48
Mathematics Subject Classification ID: 47A45
Mathematics Subject Classification ID: 47A56
zbMATH DE Number: 4165801
zbMATH Keywords: unitary colligation of operators in Krein spaces
zbMATH Keywords: characteristic function
author: Stefania A. M. Marcantognini
MaRDI profile type: Publication
cites work: Q3676678
cites work: Extensions of J-isometric and J-symmetric operators
cites work: Models for noncontractions
cites work: Q4774569
cites work: Q5658647
cites work: Q3828567
cites work: Q5612127
cites work: Q3764755
cites work: Q3879782
cites work: Q5600360
Recommended article: Q4726920
Recommended article: Q3828567
Recommended article: Q3764755
Recommended article: Q5245469
Recommended article: Q3752976
Recommended article: Unitary colligations in \(\pi _ k\)-spaces, characteristic functions and Shtraus extensions
Recommended article: Canonical forms of unbounded unitary operators in Krein spaces
Recommended article: Unitary invariants in multivariable operator theory
Recommended article: Representations of unitary relations between Kreĭn spaces
description: scientific article; zbMATH DE number 4165801