A commutant lifting theorem for a domain in \(\mathbb{C}^2\) and spectral interpolation
type: Item
instance of: scholarly article
title: A commutant lifting theorem for a domain in \(\mathbb{C}^2\) and spectral interpolation
zbMATH Open document ID: 0943.47005
author: N. J. Young
author: Jim Agler
published in: Journal of Functional Analysis
publication date: 1999-05-18T00:00:00Z
review text: The version of the commutant lifting theorem which the authors extend is Ando's theorem: any commuting pair of contractions has a dilation consisting of a commuting pair of unitaries. Note that the joint spectrum of the pair of unitaries is the two dimensional torus which serves as the distinguished boundary of the bidisk. To state the main result, consider the set of complex pairs \(\Gamma_0=\{(a+b,ab):|a|\leq 1,|b|\leq 1\}\) and its closure \(\Gamma\). Let \((S,P)\) be a pair of commuting operators on a Hilbert space. Then the authors show that this pair has a commuting normal dilation \((S',T')\) with spectrum in the distinguished boundary of \(\Gamma\) if and only if \(\Gamma\) is a complete spectral set for \((S,T)\), and if and only if certain positivity conditions are satisfied. In addition, the pair \((S',T')\) can be chosen of the form \((U+V,UV)\), where \(U\) and \(V\) are commuting unitary operators. The main result is also applied to an interpolation problem for \(2\times 2\) matrix-valued functions in the unit disk, such that the eigenvalues of the matrices are always in the closed unit disk.
reviewed by: H. Bercovici
Mathematics Subject Classification ID: 47A20
Mathematics Subject Classification ID: 47A57
zbMATH DE Number: 1270855
zbMATH Keywords: joint spectrum
zbMATH Keywords: complete spectral set
zbMATH Keywords: interpolation problem for \(2\times 2\) matrix-valued functions
describes a project that uses: Matlab
MaRDI profile type: Publication
full work available at URL: jfan.1998.3362
OpenAlex ID: W2031442788
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DOI: 10.1006/JFAN.1998.3362
description: scientific article; zbMATH DE number 1270855