Operator-valued inner product and operator inequalities
type: Item
instance of: scholarly article
title: Operator-valued inner product and operator inequalities
zbMATH Open document ID: 1151.47024
author: Jun-Ichi Fujii
published in: Banach Journal of Mathematical Analysis
publication date: 2008-05-27T00:00:00Z
full work available at URL: https://eudml.org/doc/54812
full work available at URL: v2n2.html
review text: For a Hilbert space \(H\), the \(C^*\)-inner product \({\mathbf Y}^*{\mathbf X}=\sum_{j=1}^nY_j^*X_j\) \(({\mathbf X}=(X_j)\), \({\mathbf Y}=(Y_j) \in B(H,H^n))\) makes \(B(H,H^n)\) into a Hilbert \(C^*\)-module over \(B(H)\). Using this operator-valued inner product, the author of present paper nicely presents operator versions for the Schwarz and Jensen inequalities and gives simple conditions that the equalities hold. Among his results, we mention the following version of the Schwarz inequality: for operators \(X\) and \(Y\) acting on a Hilbert space, the inequality \[ Y^*X^*YX+X^*Y^*XY\leq (XY)^*XY+(YX)^*YX \] holds with equality only when \(X\) commutes with \(Y\).
reviewed by: Mohammad Sal Moslehian
Mathematics Subject Classification ID: 47A63
Mathematics Subject Classification ID: 47A75
Mathematics Subject Classification ID: 47A80
Mathematics Subject Classification ID: 47A56
zbMATH DE Number: 5279269
zbMATH Keywords: Schwarz inequality
zbMATH Keywords: Jensen inequality
zbMATH Keywords: operator inequality
zbMATH Keywords: geometric mean
MaRDI profile type: Publication
OpenAlex ID: W2124663616
DOI: 10.15352/BJMA/1240336274
description: scientific article; zbMATH DE number 5279269