Dilatations des commutants d'opérateurs pour des espaces de Krein de fonctions analytiques. (Dilations of commutants of operators for Krein spaces of analytic functions)
type: Item
instance of: scholarly article
title: Dilatations des commutants d'opérateurs pour des espaces de Krein de fonctions analytiques. (Dilations of commutants of operators for Krein spaces of analytic functions)
zbMATH Open document ID: 0674.46019
author: Daniel Alpay
publication date: 1989-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/74857
review text: Let \({\mathcal K}_ 1\) and \({\mathcal K}_ 2\) be two Krein spaces of functions analytic in the unit disk and invariant for the left shift operator \(R_ 0\) \((R_ 0f(z)=(f(z)-f(0))/z)\), and let A be a linear continuous operator from \({\mathcal K}_ 1\) into \({\mathcal K}_ 2\) whose adjoint commutes with \(R_ 0\). We study dilations of A which preserve this commuting property and such that the Hermitian forms defined by \(I-AA^*\) and \(I- BB^*\) have the same number of negative squares. We thus obtain a version of the commutant lifting theorem in the framework of Krein spaces of analytic functions. To prove this result we suppose that the graph of the operator \(A^*\), in the metric defined by \(I-AA^*\), is a reproducing kernel Pontryagin space of analytic functions whose reproducing kernel is of the form \((J-\Theta (z)J\Theta^*(\omega))/(1- z\omega^*)\) where J is a matrix subject to \(J=J^*=J^{-1}\) and where \(\Theta\) is analytic in the unit disk and J-unitary on the unit circle.
reviewed by: Daniel Alpay
Mathematics Subject Classification ID: 46C20
Mathematics Subject Classification ID: 46E20
Mathematics Subject Classification ID: 47A20
zbMATH DE Number: 4103732
zbMATH Keywords: invariant for the left shift operator
zbMATH Keywords: dilations
zbMATH Keywords: commutant lifting theorem in the framework of Krein spaces of analytic functions
zbMATH Keywords: reproducing kernel Pontryagin space of analytic functions
MaRDI profile type: Publication
OpenAlex ID: W2321142301
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DOI: 10.5802/AIF.1200
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published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 4103732