Linear and Multilinear Algebra, ( ISI ), Volume (63), No (10), Year (2015-10) , Pages (1972-1980)

Title : ( Reverses and variations of Heinz inequality )

Authors: Mojtaba Bakherad , Mohammad Sal Moslehian ,

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Let $A, B$ be positive definite $n\times n$ matrices. We present several reverse Heinz type inequalities, in particular \begin{align*} \|AX+XB\|_2^2+ 2(\nu-1) \|AX-XB\|_2^2\leq \|A^{\nu}XB^{1-\nu}+A^{1-\nu}XB^{\nu}\|_2^2, \end{align*} where $X$ is an arbitrary $n \times n$ matrix, $\|\cdot\|_2$ is Hilbert-Schmidt norm and $\nu>1$. We also establish a Heinz type inequality involving the Hadamard product of the form \begin{align*} 2|||A^{1\over2}\circ B^{1\over2}|||\leq|||A^{s}\circ ^{1-t}+A^{1-s}\circ B^{t}||| \leq\max\{|||(A+B)\circ I|||,|||(A\circ B)+I|||\},\end{align*} in which $s, t\in [0,1]$ and $|||\cdot|||$ is a unitarily invariant norm.


, Heinz inequality, Hilbert-Schmidt norm, operator mean, Hadamard product
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author = {Bakherad, Mojtaba and Sal Moslehian, Mohammad},
title = {Reverses and variations of Heinz inequality},
journal = {Linear and Multilinear Algebra},
year = {2015},
volume = {63},
number = {10},
month = {October},
issn = {0308-1087},
pages = {1972--1980},
numpages = {8},
keywords = {Heinz inequality; Hilbert-Schmidt norm; operator mean; Hadamard product},


%0 Journal Article
%T Reverses and variations of Heinz inequality
%A Bakherad, Mojtaba
%A Sal Moslehian, Mohammad
%J Linear and Multilinear Algebra
%@ 0308-1087
%D 2015