Inequalities for C-S seminorms and Lieb functions
type: Item
instance of: scholarly article
title: Inequalities for C-S seminorms and Lieb functions
zbMATH Open document ID: 0932.15015
DOI: 10.1016/S0024-3795(98)10245-8
author: Roger A. Horn
author: Xingzhi Zhan
published in: Linear Algebra and its Applications
publication date: 2000-03-02T00:00:00Z
review text: Let \(M_n\) denote the space of \(n\times n\) complex matrices. A seminorm \(\|\cdot \|\) is called a Cauchy-Schwarz (C-S) seminorm provided \(\|A^\ast A\|=\|AA^\ast\|\) for all \(A\in M_n\) and \(\|A\|\leq \|B\|\) whenever \(A,B\) and \(B-A\) are positive semidefinite. The authors prove that any nontrivial C-S seminorm is unitarily invariant. This allows many known inequalities that pertain to unitarily invariant norms to be extended to C-S seminorms. For example, the authors prove the following generalization of a theorem of \textit{R. Bhatia} and \textit{C. Davis} [ibid. 223-224, 119-129 (1995; Zbl 0824.47006)]: Let \(A,B,X\in M_n\) be given with \(A,B\) positive semidefinite. Then \[ \||AXB|^r \|\leq \||A^pX|^r \|^{1/p} \||XB^q|^r \|^{1/q} \] for every C-S seminorm \(\|\cdot\|\) on \(M_n\), all \(r>0\) and all conjugate indices \(p,q\). The authors also discuss applications of the ideas to the Lieb class [cf. \textit{E. H. Lieb}, Adv. Math. 20, 174-178 (1976; Zbl 0324.15013)] which, as they show, can be viewed as a generalization of C-S seminorms.
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 15A45
zbMATH DE Number: 1331318
zbMATH Keywords: Cauchy-Schwarz seminorm
zbMATH Keywords: Lieb class
zbMATH Keywords: inequalities
zbMATH Keywords: unitarily invariant norms
reviewed by: Joseph Lakey
MaRDI profile type: Publication
cites work: Q3475886
cites work: A Cauchy-Schwarz inequality for operators with applications
cites work: Q4040931
cites work: An Analog of the Cauchy–Schwarz Inequality for Hadamard Products and Unitarily Invariant Norms
cites work: Cauchy-Schwarz inequalities associated with positive semidefinite matrices
cites work: Norm inequalities for fractional powers of positive operators
cites work: Inequalities for some operator and matrix functions
cites work: A representation theorem for (tr \(A^ p)^{1/p}\)
cites work: Convex and concave functions of singular values of matrix sums
cites work: Exremal Problems for Hölder Norms of Matrices and Realization of Linear Systems
cites work: Another proof of a singular value inequality concerning Hadamard products of matrices
Recommended article: On inequalities of Lieb-Thirring type
Recommended article: On an inequality of Lieb and Thirring
Recommended article: Lieb functions and \(m\)-positivity of norms
Recommended article: Q3203247
Recommended article: On inequalities for normalized Schur functions
Recommended article: On General Cwikel–Lieb–Rozenblum and Lieb–Thirring Inequalities
Recommended article: Norm inequalities for holomorphic semigroups
Recommended article: Lieb-Thirring inequality for \(L_p\) norms
Recommended article: Q4892547
Recommended article: Q4713396
description: scientific article; zbMATH DE number 1331318