Hölder-type inequalities involving unitarily invariant norms
type: Item
instance of: scholarly article
title: Hölder-type inequalities involving unitarily invariant norms
zbMATH Open document ID: 1269.47010
author: Hussien Albadawi
published in: Positivity
publication date: 2013-01-03T00:00:00Z
review text: The author proves that, if \(A, B\) and \(X\) are operators acting on a complex Hilbert space, then \[ \left| \left| \left| {} \left| A^{\ast }XB\right|^{r} \right| \right| \right| ^{2}\leq \left| \left| \left| \left( A^{\ast }\left| X^{\ast} \right| A\right) ^{\frac{ pr}{2}} \right| \right| \right| ^{\frac{1}{p}} \left| \left| \left| \left( B^{\ast }\left| X\right| B\right) ^{ \frac{qr}{2}} \right| \right| \right| ^{\frac{1}{q}} \] for all positive real numbers \(r, p\) and \(q\) such that \(p ^{ - 1} + q ^{ - 1} = 1\) and for all unitarily invariant norms. This extends some inequalities of \textit{R. Bhatia} and \textit{C. A. Davis} [Linear Algebra Appl. 223--224, 119--129 (1995; Zbl 0824.47006)] and \textit{R. A. Horn} and \textit{X.-Z. Zhan} [Linear Algebra Appl. 291, No. 1--3, 103--113 (1999; Zbl 0932.15015)]. The author also obtains general Hölder-type inequalities involving unitarily invariant norms for sums and products of Hilbert space operators by utilizing block matrices of operators, majorization of the singular values, and other classical inequalities. Relations between these inequalities and the different forms of the Cauchy-Schwarz inequalities for operators are also obtained.
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: 6121581
zbMATH Keywords: unitarily invariant norm
zbMATH Keywords: norm inequality
zbMATH Keywords: \(n\)-tuple of operators
zbMATH Keywords: Hölder's inequality
zbMATH Keywords: Cauchy-Schwarz inequality
zbMATH Keywords: positive operator
MaRDI profile type: Publication
full work available at URL: s11117-011-0122-z
OpenAlex ID: W1982904683
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DOI: 10.1007/S11117-011-0122-Z
description: scientific article; zbMATH DE number 6121581