Linear Algebra and its Applications, ( ISI ), Volume (447), No (4), Year (2014-4) , Pages (26-37)

Title : ( Bettino )

Authors: Rupinderjit Kaur , Mohammad Sal Moslehian , Mandeep Singh , Cristian Conde ,

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The celebrated Heinz inequality asserts that $ 2|||A^{1/2}XB^{1/2}|||\leq |||A^{\nu}XB^{1-\nu}+A^{1-\nu}XB^{\nu}|||\leq |||AX+XB|||$ for $X \in \mathbb{B}(\mathscr{H})$, $A,B\in \+$, every unitarily invariant norm $|||\cdot|||$ and $\nu \in [0,1]$. In this paper, we present several improvement of the Heinz inequality by using the convexity of the function $F(\nu)=|||A^{\nu}XB^{1-\nu}+A^{1-\nu}XB^{\nu}|||$, some integration techniques and various refinements of the Hermite--Hadamard inequality. In the setting of matrices we prove that \begin{eqnarray*} &&\hspace{-0.5cm}\left|\left|\left|A^{\frac{\alpha+\beta}{2}}XB^{1-\frac{\alpha+\beta}{2}}+A^{1-\frac{\alpha+\beta}{2}}XB^{\frac{\alpha+\beta}{2}}\right|\right|\right|\leq\frac{1}{|\beta-\alpha|} \left|\left|\left|\int_{\alpha}^{\beta}\left(A^{\nu}XB^{1-\nu}+A^{1-\nu}XB^{\nu}\right)d\nu\right|\right|\right|\nonumber\\ &&\qquad\qquad\leq \frac{1}{2}\left|\left|\left|A^{\alpha}XB^{1-\alpha}+A^{1-\alpha}XB^{\alpha}+A^{\beta}XB^{1-\beta}+A^{1-\beta}XB^{\beta}\right|\right|\right|\,, \end{eqnarray*} for real numbers α, β.


, Heinz inequality; convex function; Hermite, , Hadamard inequality; positive definite matrix; unitarily invariant norm
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author = {Rupinderjit Kaur and Sal Moslehian, Mohammad and Mandeep Singh and Cristian Conde},
title = {Bettino},
journal = {Linear Algebra and its Applications},
year = {2014},
volume = {447},
number = {4},
month = {April},
issn = {0024-3795},
pages = {26--37},
numpages = {11},
keywords = {Heinz inequality; convex function; Hermite--Hadamard inequality; positive definite matrix; unitarily invariant norm},


%0 Journal Article
%T Bettino
%A Rupinderjit Kaur
%A Sal Moslehian, Mohammad
%A Mandeep Singh
%A Cristian Conde
%J Linear Algebra and its Applications
%@ 0024-3795
%D 2014