Applications of the Arithmetic-Geometric and Holder Inequalities for Unitarily Invariant Norms

Abstract

Let \(A_{i},B_{i},X_{i}\) and \(Y_{i}\) be \(n\times n\) complex matrices such that \(X_{i}\) and \(Y_{i}\) are positive, \(i=1,2,...,n\), \(p,q>1\) where \(\frac{1}{p}+\frac{1}{q}=1\), \(\alpha \in \left[ 0,1\right] \) and \(r\geq 0\). Then \[ \left \vert \left \vert \left \vert \left \vert \sum \limits_{i=1}^{n}A_{i}X_{i}^{1/2}Y_{i}^{1/2}B_{i}^{\ast }\right \vert ^{r}\right \vert \right \vert \right \vert \leq \left \vert \left \vert \left \vert \left( M(\alpha )\right) ^{\frac{rp}{2}}\right \vert \right \vert \right \vert ^{1/p}\left \vert \left \vert \left \vert \left( M(1-\alpha )\right) ^{\frac{rq}{2}}\right \vert \right \vert \right \vert ^{1/q}, \] where \[M(\alpha )=\left \vert \left[ \begin{array}{cccc} \sqrt{\alpha }A_{1}X_{1}^{1/2} & \sqrt{\alpha }A_{2}X_{2}^{1/2} & \cdots & \sqrt{\alpha }A_{n}X_{n}^{1/2} \\ \sqrt{1-\alpha }B_{1}Y_{1}^{1/2} & \sqrt{1-\alpha }B_{2}Y_{2}^{1/2} & \cdots & \sqrt{1-\alpha }B_{n}Y_{n}^{1/2} \end{array} \right] \right \vert ^{2}.\] Several new results follow as special cases of this general inequality.