Löwner inequality of indefinite type
type: Item
instance of: scholarly article
title: Löwner inequality of indefinite type
zbMATH Open document ID: 1059.15025
DOI: 10.1016/S0024-3795(03)00496-8
author: Yanyan Li
published in: Linear Algebra and its Applications
publication date: 2004-08-06T00:00:00Z
review text: The author generalizes a well-known theorem of Löwner which states that for any two selfadjoint matrices \(A\), \(B\), \(A\geq B\geq 0\Rightarrow A^{1/2}\geq B^{1/2}\). A complex selfadjoint involution \(J\) defines an inner product as follows: \([x,y]:=\langle Jx,y\rangle\). Then a matrix \(A\) is called \(J\)-selfadjoint if \(JA\) is selfadjoint, and for two \(J\)-selfadjoint matrices \(A\), \(B\) the \(J\)-order relation is defined as follows: \(A\geq_JB\) if \([Ax,x]\geq [Bx,x]\) for all \(x\), or \(JA\geq JB\). Furthermore, a continuous real-valued function \(f\) on an interval \(I\) is said to be operator-monotone on \(I\) if the following property holds for all \(n\): for any two selfadjoint \(n\times n\) matrices \(A\), \(B\) all of whose eigenvalues are real and lie in \(I\), \(A\geq B\Rightarrow f(A)\geq f(B)\). Then Löwner's theorem states that the square-root function is operator-monotone on \((0,\infty)\). The main result of the paper is that for any two \(J\)-selfadjoint matrices \(A\), \(B\) all of whose eigenvalues are real and lie in \(I\), \(A\geq_J B\Rightarrow f(A)\geq_J f(B)\) for all operator-monotone functions \(f\) on \(I\). The author also discusses some corollaries involving the \(J\)-modulus of a square matrix.
Mathematics Subject Classification ID: 15A45
Mathematics Subject Classification ID: 15A63
zbMATH DE Number: 2083508
zbMATH Keywords: Löwner inequality
zbMATH Keywords: Operator monotone function
zbMATH Keywords: indefinite inner product
zbMATH Keywords: selfadjoint matrices
zbMATH Keywords: eigenvalues
reviewed by: Rabe von Randow
MaRDI profile type: Publication
cites work: Q3746364
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description: scientific article; zbMATH DE number 2083508