Linear maps compressing the polynomial numerical hulls of matrices
type: Item
instance of: scholarly article
title: Linear maps compressing the polynomial numerical hulls of matrices
zbMATH Open document ID: 1486.15034
author: Constantin Costara
published in: Linear Algebra and its Applications
publication date: 2022-01-24T00:00:00Z
review text: Let \(n\geq 2\) be an integer and let \(\mathcal{M}_{n}\) be the algebra of all complex \(n\times n\) matrices. By \(T^{t}\in \mathcal{M}_{n}\) we denote the transpose and by \(T^{\ast }\in \mathcal{M}_{n}\) the conjugate transpose of \(T\in \mathcal{M}_{n}\). Let \(\sigma (T)\) denote the spectrum of \(T\in \mathcal{M}_{n}\). For \(T\in \mathcal{M}_{n}\), we define its numerical range as \[ w(T)=\left\{ x^{\ast }Tx:x\in \mathbb{C}^{n},\left\Vert x\right\Vert =1\right\} . \] Let \(\mathcal{P}_{k}\) be the set of all polynomials with complex coefficients of degree at most \(k\) and let \(\left\Vert .\right\Vert\) denote the operator norm in \(\mathcal{M}_{n}\). The polynomial numerical hull of order \(k\geq 1\) of \(T\in \mathcal{M}_{n}\) is defined as \[ V^{k}(T)=\left\{ \lambda \in \mathbb{C}:\left\vert p(\lambda )\right\vert \leq \left\Vert p(T)\right\Vert \text{ for every }p\in \mathcal{P}_{k}\right\} . \] Note that the concept of polynomial numerical hull extends the notion of numerical range of \(T\in \mathcal{M}_{n}\), namely \(V^{1}(T)=w(T)\). In [Linear Algebra Appl. 575, 27--34 (2019; Zbl 1416.15021)], \textit{Gh. Aghamollaei} et al. characterized linear maps on \(\mathcal{M}_{n}\) which preserve the polynomial numerical hull of matrices of some fixed order \(k\). It turns out that for a linear map \(\Phi : \mathcal{M}_{n}\rightarrow \mathcal{M}_{n}\) and an integer \(k\) with \(1\leq k<n\) we have \[ V^{k}\left( \Phi (T)\right) =V^{k}(T)\quad \text{for every }T\in \mathcal{M}_{n}, \] if and only if there exists a unitary matrix \(U\in \mathcal{M}_{n}\) such that either \[ \Phi (T)=UTU^{\ast }\quad \text{for every }T\in \mathcal{M}_{n} \tag{1} \] or \[ \Phi (T)=UT^{t}U^{\ast }\quad \text{for every }T\in \mathcal{M}_{n}. \tag{2} \] In the present paper, the author studies linear maps \(\Phi\) on \(\mathcal{M}_{n}\) which compress the polynomial numerical hull of order \(k\), i.e., \[ V^{k}\left( \Phi (T)\right) \subseteq V^{k}(T)\quad \text{for every }T\in \…
reviewed by: Janko Marovt
Mathematics Subject Classification ID: 15A86
Mathematics Subject Classification ID: 15A04
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 47A12
zbMATH DE Number: 7462787
zbMATH Keywords: linear preserver
zbMATH Keywords: matrix algebra
zbMATH Keywords: numerical range
zbMATH Keywords: polynomial numerical hull
MaRDI profile type: Publication
full work available at URL: j.laa.2021.12.004
OpenAlex ID: W4200164338
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DOI: 10.1016/J.LAA.2021.12.004
description: scientific article; zbMATH DE number 7462787