Some operator equalities involving connections and means
type: Item
instance of: scholarly article
title: Some operator equalities involving connections and means
zbMATH Open document ID: 0912.47010
DOI: 10.1016/S0024-3795(96)00285-6
author: Jaspal Singh Aujila
published in: Linear Algebra and its Applications
publication date: 1999-02-10T00:00:00Z
review text: Given a positive operator monotone function \(f\), one can define a binary operation \((A,B)\mapsto A\sigma B\) on the set of all positive bounded operators on a Hilbert space by setting \(A\sigma B=A^{1/2}f(A^{-1/2}BA^{-1/2})A^{1/2}\). These operations were called connections by A. Kubo and T. Ando. Given a connection \(\sigma\), one can construct several related connections. For instance, \(A\sigma^\perp B=(A^{-1}\sigma B^{-1})^{-1}\). Theorem 2.1. Fix connections \(\tau,\nu\). The identity \((A\tau B)(A\sigma B)^{-1}(A\nu B)=A\sigma^\perp B\) holds for all \(A,B\) and \(\sigma\) if and only if \(\nu=\tau^\perp\). Another result of this nature is proved, and consequences deduced.
Mathematics Subject Classification ID: 47A63
zbMATH DE Number: 1044235
zbMATH Keywords: operator monotone function
zbMATH Keywords: operator mean
zbMATH Keywords: operator connection
reviewed by: H. Bercovici
MaRDI profile type: Publication
cites work: Series and parallel addition of matrices
cites work: Q4169818
cites work: Q3874790
cites work: Means of positive linear operators
cites work: Characterization of operations derived from network connections
Wikidata QID: Q127220111
full work available at URL: s0024-3795(96)00285-6
OpenAlex ID: W1985664285
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description: scientific article; zbMATH DE number 1044235