Inequalities for sums and direct sums of Hilbert space operators
type: Item
instance of: scholarly article
title: Inequalities for sums and direct sums of Hilbert space operators
zbMATH Open document ID: 1116.47012
author: Omar Hirzallah
author: Fuad Kittaneh
published in: Linear Algebra and its Applications
publication date: 2007-06-06T00:00:00Z
review text: The authors investigate several singular value inequalities and norm inequalities involving sums and direct sums of operators. They prove the following nice Cauchy--Schwarz type inequality for singular values: \[ s_j \left(\sum_{i=1}^n A_iX_iB_i\right) \leq \left\| \sum_{i=1}^n| A_i^*| ^2\right\| ^{1/2} \left\| \sum_{i=1}^n| B_i| ^2\right\| ^{1/2} s_j\left(\oplus_{i=1}^n X_i\right),\quad j=1, 2, \dots, \] where \(A_i, B_i, X_i\), \(1\leq i\leq n\), are bounded linear operators on a Hilbert space and the operators \(X_i\), \(1\leq i\leq n\), are compact. They also show that if \(X\) and \(Y\) are compact operators on a Hilbert space, then \(s_j\left(\frac{X + Y}{2}\right)\leq s_j(X \oplus Y)\), \(j=1, 2, \dots\). The authors generalize the inequality \[ \left|\left|\left|\left(\frac{X+Y}{2}\right)\oplus\left(\frac{X+Y}{2}\right)\right|\right|\right|\leq||| X\oplus Y|||, \] where \(|||\cdot|||\) is a unitarily invariant norm and \(X\) and \(Y\) are bounded linear operators on a Hilbert space by giving a weighted mean norm inequality.
reviewed by: Mohammad Sal Moslehian
Mathematics Subject Classification ID: 47A30
Mathematics Subject Classification ID: 47A63
Mathematics Subject Classification ID: 47B10
Mathematics Subject Classification ID: 47B15
zbMATH DE Number: 5161805
zbMATH Keywords: compact operator
zbMATH Keywords: singular value
zbMATH Keywords: unitarily invariant norm
zbMATH Keywords: inequality
MaRDI profile type: Publication
full work available at URL: j.laa.2006.03.036
OpenAlex ID: W2078226957
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DOI: 10.1016/J.LAA.2006.03.036
description: scientific article; zbMATH DE number 5161805