Young type inequalities for positive operators
type: Item
instance of: scholarly article
title: Young type inequalities for positive operators
zbMATH Open document ID: 1279.47031
author: Cristian Conde
published in: Annals of Functional Analysis
publication date: 2013-04-12T00:00:00Z
full work available at URL: AFA-tex v4 n2 a12.pdf
review text: The well-known Young inequality states that, if \(a,b\geq0\) and \(0\leq\nu\leq1\), then \[ a^{\nu}b^{1-\nu}\leq \nu a+(1-\nu)b. \] It is known that, if we replace the scalars \(a,b\) with two positive operators \(A,B\in\mathbb{B}(H)\), the above inequality does not hold. However, \textit{T. Ando} [Oper. Theory, Adv. Appl. 75, 33--38 (1994; Zbl 0830.47010)] proved that \[ |||A^{\nu}X B^{1-\nu}|||\leq \nu |||AX|||+(1-\nu)|||XB||| \] for positive operators \(A,B\) and a unitarily invariant norm \(|||\cdot|||\). In Section 2, the author presents a refinement of Ando's inequality. In Section 3, he reviews some results related to the Heinz inequality [\textit{E. Heinz}, Math. Ann. 123, 415--438 (1951; Zbl 0043.32603)] and improves some of them. Furthermore, some new proofs of known results are obtained.
reviewed by: Maryam Khosravi
Mathematics Subject Classification ID: 47A63
Mathematics Subject Classification ID: 47A30
Mathematics Subject Classification ID: 15A60
zbMATH DE Number: 6153500
zbMATH Keywords: Young inequality
zbMATH Keywords: Heinz inequality
zbMATH Keywords: positive operators
zbMATH Keywords: unitarily invariant norm
MaRDI profile type: Publication
OpenAlex ID: W1967133834
DOI: 10.15352/AFA/1399899532
description: scientific article; zbMATH DE number 6153500