Classification of hereditary matrices
type: Item
instance of: scholarly article
title: Classification of hereditary matrices
zbMATH Open document ID: 0939.47014
DOI: 10.1016/S0024-3795(97)00307-8
author: Jim Agler
author: J. William Helton
author: Mark Stankus
published in: Linear Algebra and its Applications
publication date: 2000-07-05T00:00:00Z
review text: Let \({\mathcal H}\) be a complex Hilbert space. For a polynomial \(p\in\mathbb{C}[x, y]\), \(p(x,y)= \sum p_{mn}y^nx^m\), and an operator \(T\in{\mathcal L}({\mathcal H})\) the authors define \(p(T)= \sum p_{mn}T^{*n} T^m\). \(T\) is called a root of \(p\) if \(p(T)= 0\). For the case that \(T\) is an algebraic operator, the authors investigate the relationships which \(p(T)= 0\) forces between \(p\) and \(T\). This relationship is in terms of invariant subspaces of \(T\) and vanishing of coefficients of \(p\), leading to certain graphs. For diagonalizable operators \(T\) and for operators with exactly one or two eigenvalues, the authors characterize \(p(T)= 0\) by graph theoretic conditions.
Mathematics Subject Classification ID: 47A65
Mathematics Subject Classification ID: 47A15
Mathematics Subject Classification ID: 47A60
zbMATH DE Number: 1154659
zbMATH Keywords: root
zbMATH Keywords: algebraic operator
zbMATH Keywords: invariant subspaces
zbMATH Keywords: diagonalizable operators
zbMATH Keywords: eigenvalues
reviewed by: Jürgen Voigt
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1154659