Superadditivity and derivative of operator functions
type: Item
instance of: scholarly article
title: Superadditivity and derivative of operator functions
zbMATH Open document ID: 1315.47014
author: Mitsuru Uchiyama
author: Atsushi Uchiyama
author: Mariko Giga
published in: Linear Algebra and its Applications
publication date: 2014-11-17T00:00:00Z
review text: \textit{M. S. Moslehian} and \textit{H. Najafi} [Integral Equations Oper. Theory 71, No. 4, 575--582 (2011; Zbl 1272.47026)] showed that, if \(A\) and \(B\) are positive operators, then \(AB+BA\geq 0\) if and only if \(f(A+B)\leq f(A)+f(B)\) for every operator monotone function \(f\) on \([0,\infty)\). In the present work, this result is generalized by showing that, if \(A_i\geq 0\) \((i=1,\dots,n)\), then \(\sum_{i\neq j}A_iA_j\geq 0\) if and only if \(f(A_1+\dots+A_n)\leq f(A_1)+\cdots + f(A_n)\) for every non-negative operator monotone function \(f\) on \([0,\infty)\). Moreover, the authors present some other conditions equivalent to the positivity of \(AB+BA\) using the Fréchet derivatives of operator convex functions.
reviewed by: Mohsen Kian
Mathematics Subject Classification ID: 47A56
Mathematics Subject Classification ID: 47A63
zbMATH DE Number: 6370320
zbMATH Keywords: matrix order
zbMATH Keywords: Fréchet derivative
zbMATH Keywords: operator monotone function
zbMATH Keywords: operator concave function
zbMATH Keywords: operator convex function
zbMATH Keywords: positive operator
MaRDI profile type: Publication
full work available at URL: j.laa.2014.09.006
OpenAlex ID: W2069275387
cites work: Norm inequalities related to operator monotone functions
cites work: Q5690490
cites work: A matrix subadditivity inequality for \(f(A + B)\) and \(f(A) + f(B)\)
cites work: A Schwarz inequality for positive linear maps on C\(^*\)-algebras
cites work: A Schwarz Inequality for Convex Operator Functions
cites work: Jensen's inequality for operators and Loewner's theorem
cites work: Loewner matrices of matrix convex and monotone functions
cites work: Q4040931
cites work: Around operator monotone functions
cites work: Commutativity of selfadjoint operators
cites work: Powers and commutativity of selfadjoint operators
cites work: Inverse functions of polynomials and orthogonal polynomials as operator monotone functions
cites work: Q3603790
cites work: Subadditivity of eigenvalue sums
cites work: Operator monotone functions, positive definite kernels and majorization
DOI: 10.1016/J.LAA.2014.09.006
description: scientific article; zbMATH DE number 6370320