Mathematische Zeitschrift, Year (2017-1) , Pages (1-10)

Title : ( Operator means and positivity of block operators )

Authors: Hamed Najafi ,

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Abstract

‎The aim of this paper is to find some sufficient conditions for positivity of block matrices of positive operators‎. ‎It is shown that for positive operators $A,B,C$ and for every non-negative operator monotone function $f$ on $‎ ‎(0,\infty)$‎, ‎the block matrix‎ ‎\begin{align*}‎ ‎\left(‎ ‎\begin{array}{cc}‎ ‎f(A) & f(C) \\‎ ‎f(C) & f(B) \\‎ ‎\end{array}‎ ‎\right)‎ ‎\end{align*}‎ ‎is positive if and only if $C \leq A!B$‎. ‎In particular‎, ‎if $C \leq A!B$ then‎ ‎\begin{equation*}‎ ‎\left(‎ ‎\begin{array}{cc}‎ ‎A & C \\‎ ‎C & B \\‎ ‎\end{array}‎ ‎\right)‎, ‎\end{equation*}‎ ‎is positive‎.

Keywords

, Positive block matrix, Operator monotone function, Operator mean
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@article{paperid:1066359,
author = {Najafi, Hamed},
title = {Operator means and positivity of block operators},
journal = {Mathematische Zeitschrift},
year = {2017},
month = {January},
issn = {0025-5874},
pages = {1--10},
numpages = {9},
keywords = {Positive block matrix-Operator monotone function-Operator mean},
}

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%0 Journal Article
%T Operator means and positivity of block operators
%A Najafi, Hamed
%J Mathematische Zeitschrift
%@ 0025-5874
%D 2017

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