Equality of three numerical radius inequalities
type: Item
instance of: scholarly article
title: Equality of three numerical radius inequalities
zbMATH Open document ID: 1410.15044
author: Hwa-Long Gau
author: Pei Yuan Wu
published in: Linear Algebra and its Applications
publication date: 2018-08-01T00:00:00Z
review text: Let $A$ be an $n\times n$ matrix over the field of complex numbers and let $w(A)$ and $\| A\| $ numerical radius and operator norm respectively. The following inequalities are well-known: \[ w(A)\le\| A\|, \] \[ w(A)^2\le\frac{\| A\| ^2 +w(A^2)}{2}, \] \[ w(A)\le\frac{ \| A\| +\| A^2\| ^{1/2}}{2}, \] \[ w(A)^2\le\frac{ \| A\| + w(\Delta _t(A))}{2}, \] with $\Delta_t(A)=|A|^tV|A|^{1-t}$, $0\le t\le 1$, being the generalized Aluthge transform of $A$ where $|A|=(A^*A)^{1/2}$ and $V$ is the partial isometry defined on $\mathcal{C}^n=\operatorname{ran }A^*\oplus\operatorname{ker }A$ by $V(|A|x)=Ax$ for any vector $x$ in $\mathcal{C}^n$. In this paper the authors, by imposing certain conditions on matrix $A$, prove equalities for the last three inequalities. More specifically, they prove that \[ w(A)^2=\frac{ \| A\| ^2 +w(A^2)}{2}, \] \[ w(A)=\frac{ \| A\| +\| A^2\| ^{1/2}}{2}, \] \[ w(A)^2=\frac{ \| A\| + w(\Delta _t(A))}{2}. \] They also remark that these three equalities are true for bounded linear operators on a Hilbert space.
reviewed by: Jaspal Singh Aujla
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 47A12
zbMATH DE Number: 6912318
zbMATH Keywords: numerical range
zbMATH Keywords: numerical radius
zbMATH Keywords: operator norm
zbMATH Keywords: generalized Aluthge transform
MaRDI profile type: Publication
full work available at URL: j.laa.2018.05.021
OpenAlex ID: W2803482533
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Wikidata QID: Q129782090
DOI: 10.1016/J.LAA.2018.05.021
description: scientific article; zbMATH DE number 6912318