Some inequalities for the Euclidean operator radius of two operators in Hilbert spaces
type: Item
instance of: scholarly article
title: Some inequalities for the Euclidean operator radius of two operators in Hilbert spaces
zbMATH Open document ID: 1112.47001
published in: Linear Algebra and its Applications
publication date: 2006-12-07T00:00:00Z
review text: The Euclidean operator radius of a pair of bounded linear operators \(A\) and \(B\) on a Hilbert space \(H\) with inner product \(\langle\cdot,\cdot\rangle\) is \(w_e(A, B)= \sup_{\| x\|= 1}(|\langle Ax, x\rangle|^2+ |\langle Bx, x\rangle|^2)^{1/2}\), as was first defined by \textit{G.\,Popescu} [``Unitary invariants in multivariable operator theory'' (Preprint) (2004; arXiv:math/0410492)]. For selfadjoint \(A\) and \(B\), \(W_e(A, B)\) reduces to the numerical radius \(w(A+ iB)\) of \(A+ iB\). In the present paper, the author proves miscellaneous sharp inequalities involving the Euclidean operator radii, numerical radii and norms of various operators, which generalize some previous ones by \textit{F.\,Kittaneh} [Stud.\ Math.\ 168, No.\,1, 73--80 (2005; Zbl 1072.47004)] and Popescu. For example, it is shown that \(w_e(A, B)\geq (\sqrt{2}/2) w(A^2+ B^2)^{1/2}\) and, as a consequence, \(w(C)^2\geq w(\alpha^2 C^2+ \beta^2 C^{*2})/(2|\alpha|^2+ 2|\beta|^2)\) for any operator \(C\) and scalars \(\alpha\) and \(\beta\) not both zero. On the other hand, some sharp upper bounds for \(w_e(A, B)^2\) are shown to be \(\max\{\| A\|^2,\| B\|^2\}+ w(B^*A)\) and \([\| A^*A+ B^*B\|+\| A^*A- B^*B\|]/2+ w(B^* A)\) and, from the former, one for \(w(C)^2\) is \([\max\{\| C+ C^*\|^2,\| C- C^*\|^2\}+ w((C^*- C)(C+ C^*))]/4\). The proofs for these are all elementary.
Mathematics Subject Classification ID: 47A12
Mathematics Subject Classification ID: 47A30
zbMATH DE Number: 5077737
zbMATH Keywords: numerical radius
zbMATH Keywords: Euclidean radius
zbMATH Keywords: bounded linear operators
zbMATH Keywords: operator norm
zbMATH Keywords: self-adjoint operators
author: Silvestru Sever Dragomir
MaRDI profile type: Publication
full work available at URL: j.laa.2006.04.017
OpenAlex ID: W2039900885
cites work: Q3512117
cites work: Q5521075
cites work: Numerical radius inequalities for Hilbert space operators
cites work: Unitary invariants in multivariable operator theory
DOI: 10.1016/J.LAA.2006.04.017
Recommended article: Euclidean operator radius inequalities of a pair of bounded linear operators and their applications
Recommended article: Upper bounds for numerical radius inequalities involving off-diagonal operator matrices
Recommended article: Upper bounds for the Euclidean operator radius and applications
Recommended article: Extensions of Euclidean operator radius inequalities
Recommended article: Numerical radius inequalities for certain \(2\times 2\) operator matrices. II
Recommended article: INEQUALITIES FOR THE NORM AND THE NUMERICAL RADIUS OF LINEAR OPERATORS IN HILBERT SPACES
Recommended article: Numerical Radius Inequalities for Several Operators
Recommended article: More accurate numerical radius inequalities (I)
Recommended article: Numerical radius inequalities for Hilbert space operators
Recommended article: Some upper bounds for the Davis-Wielandt radius of Hilbert space operators
description: scientific article; zbMATH DE number 5077737