Matrix Hölder-McCarthy inequality via matrix geometric means
type: Item
instance of: scholarly article
title: Matrix Hölder-McCarthy inequality via matrix geometric means
zbMATH Open document ID: 1450.15019
author: Ryosuke Nakayama
author: Yuki Seo
author: Reo Tojo
published in: Advances in Operator Theory
publication date: 2020-07-27T00:00:00Z
review text: This paper examines a number of matrix inequalities which generalize well-known scalar inequalities and uses matrix geometric means to give unified proofs. If \(A\) and \(B\) are \(n\times n\) positive definite matrices then the matrix weighted geometric mean is defined by \(A\natural_{p}B:=A^{1/2}(A^{-1/2}BA^{-1/2})^{p}A^{1/2}\) for \(p\in\lbrack0,1]\). The definition can be extended in certain cases to semidefinite matrices by continuity and by using Moore-Penrose inverses to a larger range of \(p\) although \(\natural_{p}\) may not be a matrix mean in the sense of Kubo-Ando when \(p\notin\lbrack0,1]\). The authors prove the following version of the Hölder-McCarthy inequality (Theorem 1): if \(A\) is positive semidefinite and \(X\) is an \(n\times k\) matrix then \(X^{\ast}A^{p}X\leq(X^{\ast}X)\natural_{p}(X^{\ast}AX)\) for all \(p\in\lbrack0,1]\) and the reverse inequality holds for \(p\in\lbrack1,2]\). Moreover if \(A\) is positive definite then the reverse inequality also holds for \(p\in\lbrack-1,0]\). On the other hand, suppose in addition that the eigenvalues of \(A\) lie in an interval \([m,M]\) where \(0 < m < M\). Then a matrix version of the Kantorovich inequality holds (Theorem 3): \(K(h,p)(X^{\ast}X)\natural_{p}(X^{\ast}AX)\leq X^{\ast}A^{p}X\leq(X^{\ast}X)\natural_{p}(X^{\ast}AX)\) for all \(p\in\lbrack0,1],\) where \(K(h,p)\) in the Kantorovich constant which depends only on \(h:=M/m\) and \(p\). Again the reverse inequality holds when \(p\in\lbrack-1,0]\cup\lbrack1,2]\). The paper explores related inequalities and gives examples to show that some matrix generalizations of these inequalities fail.
reviewed by: John D. Dixon
Mathematics Subject Classification ID: 15A45
Mathematics Subject Classification ID: 15A10
Mathematics Subject Classification ID: 47A63
Mathematics Subject Classification ID: 47A64
zbMATH DE Number: 7225053
zbMATH Keywords: matrix inequalities
zbMATH Keywords: Hölder-McCarthy inequality
zbMATH Keywords: quasi-arithmetic power mean
zbMATH Keywords: positive definite matrix
MaRDI profile type: Publication
full work available at URL: s43036-020-00044-y
OpenAlex ID: W3007225886
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DOI: 10.1007/S43036-020-00044-Y
description: scientific article; zbMATH DE number 7225053