Some operator inequalities for unitarily invariant norms
type: Item
instance of: scholarly article
title: Some operator inequalities for unitarily invariant norms
zbMATH Open document ID: 1394.47018
DOI: 10.1215/20088752-0000009X
author: Jianguo Zhao
author: Junliang Wu
published in: Annals of Functional Analysis
publication date: 2017-03-14T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.afa/1485918117
review text: Denote by \(\mathbf{B}(\mathcal{H})\) the algebra of all bounded linear operators on a complex separable Hilbert space \(\mathcal{H}\). For \(A\in \mathbf{B}(\mathcal{H})\), \(A\geq 0\) means that \(A\) is positive and \(A > 0\) means that \(A\) is invertible and positive. Let \(A, B, X \in M_{n}\) with \(A, B > 0\). The norm inequality \[ (2 + t){| \kern -0.25ex | \kern -0.25ex | A^{r}XB^{2 - r} + A^{2 - r}XB^{r} | \kern -0.25ex | \kern -0.25ex | } \leq 2 {| \kern -0.25ex | \kern -0.25ex | A^{2}X + tAXB + XB^{2} | \kern -0.25ex | \kern -0.25ex | } \tag{1} \] holds for every unitarily invariant norm \({| \kern -0.25ex | \kern -0.25ex | \cdot | \kern -0.25ex | \kern -0.25ex | }\), \(-2< t\leq 2\) and \(\frac{1}{2}\leq r\leq \frac{3}{2}\), see [\textit{X.-Z. Zhan}, SIAM J. Matrix Anal. Appl. 20, No. 2, 466--470 (1998; Zbl 0921.15011)]. Another important norm inequality is the well-known Cauchy-Schwarz type inequality: for every unitarily invariant norm \({| \kern -0.25ex | \kern -0.25ex | \cdot | \kern -0.25ex | \kern -0.25ex | }\) and every positive real number \(r\), the function \[ g(t) = {| \kern -0.25ex | \kern -0.25ex | |A^{t}XB^{1 - t}|^{r} | \kern -0.25ex | \kern -0.25ex | } \cdot {| \kern -0.25ex | \kern -0.25ex | |A^{1 - t}XB^{t}|^{r} | \kern -0.25ex | \kern -0.25ex | } \tag{2} \] is convex on the interval \([0, 1]\) and attains its minimum at \(t = \frac{1}{2}\). Assume now that \({| \kern -0.25ex | \kern -0.25ex | \cdot | \kern -0.25ex | \kern -0.25ex | }\) is a unitarily invariant norm on a two-sided ideal \(K_{{| \kern -0.25ex | \kern -0.25ex | \cdot | \kern -0.25ex | \kern -0.25ex | }}\) in \(\mathbf{B}(\mathcal{H})\). The following inequality is a consequence of the norm inequalities of the Cauchy-Schwarz type. Let \(p\) and \(q\) be two positive real numbers with \(p > 0\) or \(q > 0\). Then \[ {| \kern -0.25ex | \kern -0.25ex | |A^{p}XB^{q}|^{r} | \kern -0.25ex | \kern -0.25ex | } \cdot {| \kern -0.25ex | \kern -0.25ex | |A^{q}XB^{p}|^{r} | \kern -0.…
reviewed by: Tin-Yau Tam
Mathematics Subject Classification ID: 47A30
Mathematics Subject Classification ID: 47A63
Mathematics Subject Classification ID: 15A60
zbMATH DE Number: 6694412
zbMATH Keywords: Zhan's inequality
zbMATH Keywords: positive operators
zbMATH Keywords: unitarily invariant norms
zbMATH Keywords: Cauchy-Schwarz inequality
MaRDI profile type: Publication
OpenAlex ID: W2583146938
description: scientific article; zbMATH DE number 6694412