Banach Journal of Mathematical Analysis

Interpolating inequalities for functions of positive semidefinite matrices

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.

Article information

Source
Banach J. Math. Anal., Volume 12, Number 4 (2018), 955-969.

Dates
Accepted: 12 March 2018
First available in Project Euclid: 10 July 2018

https://projecteuclid.org/euclid.bjma/1531209674

Digital Object Identifier
doi:10.1215/17358787-2018-0008

Mathematical Reviews number (MathSciNet)
MR3858756

Zentralblatt MATH identifier
06946298

Citation

Al-Natoor, Ahmad; Hirzallah, Omar; Kittaneh, Fuad. Interpolating inequalities for functions of positive semidefinite matrices. Banach J. Math. Anal. 12 (2018), no. 4, 955--969. doi:10.1215/17358787-2018-0008. https://projecteuclid.org/euclid.bjma/1531209674

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