Interpolating inequalities for functions of positive semidefinite matrices

Ahmad Al-Natoor; Omar Hirzallah; Fuad Kittaneh

doi:10.1215/17358787-2018-0008

October 2018 Interpolating inequalities for functions of positive semidefinite matrices

Ahmad Al-Natoor, Omar Hirzallah, Fuad Kittaneh

Banach J. Math. Anal. 12(4): 955-969 (October 2018). DOI: 10.1215/17358787-2018-0008

Abstract

Let $A$ , $B$ be positive semidefinite $n\times n$ matrices, and let $\alpha\in(0,1)$ . We show that if $f$ is an increasing submultiplicative function on $[0,\infty)$ with $f(0)=0$ such that $f(t)$ and $f^{2}(t^{1/2})$ are convex, then $\begin{eqnarray*}\big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert f(\llvert AB\rrvert )\big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert^{2}&\leq&f^{4}(\frac{1}{(4\alpha(1-\alpha))^{1/4}})(\big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert(\alpha f(A)+(1-\alpha )f(B))^{2}\big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert\\&&{}\times \big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert((1-\alpha)f(A)+\alpha f(B))^{2}\big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert)\end{eqnarray*}$ for every unitarily invariant norm. Moreover, if $\alpha\in{}[0,1]$ and $X$ is an $n\times n$ matrix with $X\neq0$ , then $\begin{eqnarray*}&&\big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert f(\llvert AXB\rrvert )\big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert^{2}\\&&\quad \leq\frac{f(\Vert X\Vert)}{\Vert X\Vert}\big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert\alpha f^{2}(A)X+(1-\alpha)Xf^{2}(B)\big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert \big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert(1-\alpha)f^{2}(A)X+\alpha Xf^{2}(B)\big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert\end{eqnarray*}$ for every unitarily invariant norm. These inequalities present generalizations of recent results of Zou and Jiang and of Audenaert.

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Ahmad Al-Natoor, Omar Hirzallah, and Fuad Kittaneh "Interpolating inequalities for functions of positive semidefinite matrices," Banach Journal of Mathematical Analysis 12(4), 955-969, (October 2018). https://doi.org/10.1215/17358787-2018-0008

Received: 23 December 2017; Accepted: 12 March 2018; Published: October 2018

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