Noncommutative multivariable operator theory
type: Item
instance of: scholarly article
title: Noncommutative multivariable operator theory
zbMATH Open document ID: 1277.47017
author: Gelu Popescu
published in: Integral Equations and Operator Theory
publication date: 2013-04-15T00:00:00Z
review text: Given a Hilbert space \(\mathcal{H}\), the author studies noncommutative domains \(\mathbb{D}_f^\phi(\mathcal{H})\) in \(B(\mathcal{H})^n\), generated by a positive regular free holomorphic function \(f\) and some classes of \(n\)-tuples \(\phi=(\phi_1,\dots,\phi_n)\) of formal power series in (noncommutative) indeterminates \(Z_1,\dots,Z_n\). Such a domain \(\mathbb{D}_f^\phi=\mathbb{D}_f^\phi(\mathcal{H})\) has a universal model associated to the multiplication operators \((M_{Z_1},\dots,M_{Z_n})\), described via noncommutative Poisson transforms. Among several properties, all joint invariant subspaces under \(M_{Z_1},\dots,M_{Z_n}\) are given a Beurling type characterization, and the Nevanlinna-Pick interpolation problem for the noncommutative Hardy algebra \(H^\infty(\mathbb{D}_f^\phi)\) is solved.
reviewed by: Florian-Horia Vasilescu
Mathematics Subject Classification ID: 47A45
Mathematics Subject Classification ID: 46L52
Mathematics Subject Classification ID: 47A20
Mathematics Subject Classification ID: 46T25
Mathematics Subject Classification ID: 46L07
zbMATH DE Number: 6153958
zbMATH Keywords: multivariable operator theory
zbMATH Keywords: noncommutative Reinhardt domain
zbMATH Keywords: free biholomorphic function
zbMATH Keywords: weighted Fock space
zbMATH Keywords: invariant subspace
zbMATH Keywords: noncommutative Poisson transform
zbMATH Keywords: characteristic function
zbMATH Keywords: operator model theory
zbMATH Keywords: commutant lifting
zbMATH Keywords: Nevanlinna-Pick interpolation
MaRDI profile type: Publication
full work available at URL: s00020-012-2012-6
OpenAlex ID: W2023369354
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DOI: 10.1007/S00020-012-2012-6
description: scientific article; zbMATH DE number 6153958