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Some extensions of the Young and Heinz inequalities for Matrices by Hajmohamadi, Monire; Lashkaripour, Rahmatollah; Bakherad, Mojtaba is a scholarly article available to read on EtoBox.

What is Some extensions of the Young and Heinz inequalities for Matrices about?

In this paper, we present some extensions of the Young and Heinz inequalities for the Hilbert-Schmidt norm as well as any unitarily invariant norm. Furthermore, we give some inequalities dealing with matrices. More precisely, for two positive semidefinite matrices $A$ and $B$ we show that \begin{align*} \Big\|A^{\nu}XB^{1-\nu}+A^{1-\nu}XB^{\nu}\Big\|_{2}^{2}\leq\Big\|AX+XB\Big\|_{2}^{2}- 2r\Big\|AX-XB\Big\|_{2}^{2}-r_{0}\left(\Big\|A^{\frac{1}{2}}XB^{\frac{1}{2}}-AX\Big\|_{2}^{2}+ \Big\|A^{\frac{1}{2}}XB^{\frac{1}{2}}-XB\Big\|_{2}^{2}\right), \end{align*} where $X$ is an arbitrary $n\times n$ matrix, $0<\nu\leq\frac{1}{2}$, $r=\min\{\nu, 1-\nu\}$ and $r_{0}=\min\{2r, 1-2r\}$.

Author

Hajmohamadi, Monire; Lashkaripour, Rahmatollah; Bakherad, Mojtaba

Published

2017

Language

EN