A Dunkl-Williams type inequality for absolute value operators
type: Item
instance of: scholarly article
title: A Dunkl-Williams type inequality for absolute value operators
zbMATH Open document ID: 1195.26044
author: Kichi-Suke Saito
author: Masaru Tominaga
published in: Linear Algebra and its Applications
publication date: 2010-05-11T00:00:00Z
review text: The Dunkl-Williams inequality states that for any nonzero elements \(x,y\) in a normed linear space \(X\) the following inequality holds: \[ \left\|\frac{x}{\|x\|}-\frac{y}{\|y\|}\right\| \leq \frac{4\|x-y\|}{\|x||+\|y\|}. \] The authors generalize this inequality considering the algebra \(B({\mathcal H})\) of all bounded linear operators acting on a complex Hilbert space \({\mathcal(H)}\). The main result is the following: Let \(A,B\in B({\mathcal H})\) be operators with polar decompositions \(A=U|A|\) and \(B=V|B|\), and let \(p,q>1\) with \(\frac{1}{p}+\frac{1}{q}=1\). Then \[ |(U-V)|A||^2\leq p|A-B|^2+q(|A|-|B|)^2. \] The equality holds iff \(p(A-B)=qV(|B|-|A|)\) and \(U^*U=V^*V.\) Also, they give a refinement of the equality conditions.
reviewed by: Sanja Varošanec
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 47A63
Mathematics Subject Classification ID: 26D99
zbMATH DE Number: 5706981
zbMATH Keywords: Dunkl-Williams inequality
zbMATH Keywords: Bohr inequality
zbMATH Keywords: operator inequality
MaRDI profile type: Publication
full work available at URL: j.laa.2010.01.016
OpenAlex ID: W1992541809
cites work: A Simple Norm Inequality
cites work: Non-commutative operator Bohr inequality
cites work: Analytic Inequalities
cites work: Q3553781
DOI: 10.1016/J.LAA.2010.01.016
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description: scientific article; zbMATH DE number 5706981