Characterizations of the polynomial numerical hull of degree \(k\)
type: Item
instance of: scholarly article
title: Characterizations of the polynomial numerical hull of degree \(k\)
zbMATH Open document ID: 1106.15018
author: James V. Burke
author: Anne Greenbaum
published in: Linear Algebra and its Applications
publication date: 2006-12-07T00:00:00Z
review text: For bounded linear operators on a Hilbert space, six characterizations of the polynomial numerical hull of degree~\(k\) are established. It is shown how these characterizations provide a natural distinction between interior and boundary points. One of the characterizations is used to prove that the polynomial numerical range of any fixed degree \(k\) for a Toeplitz matrix whose symbol is piecewise continuous and approaches all or most of the infinite-dimensional Toeplitz operators, as the matrix size goes to infinity. More precisely, it is shown that the polynomial numerical hull of degree \(k\) for the Toeplitz operator contains the uniform and partial limits as \(N\to \infty\) of the polynomial numerical hulls of degree \(k\) for the \(N\times N\) Toeplitz matrix, which in turn contain the closure of the interior of that of the Toeplitz operator.
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 47A12
Mathematics Subject Classification ID: 47B35
zbMATH DE Number: 5077725
zbMATH Keywords: polynomial numerical hull
zbMATH Keywords: nonnormal matrix
zbMATH Keywords: Toeplitz matrix
zbMATH Keywords: Hilbert space
zbMATH Keywords: Toeplitz operators
reviewed by: A. R. Alimov
describes a project that uses: Eigtool
MaRDI profile type: Publication
full work available at URL: j.laa.2006.04.023
OpenAlex ID: W2056696190
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DOI: 10.1016/J.LAA.2006.04.023
description: scientific article; zbMATH DE number 5077725