Some hybrid matrix triangle inequalities

Abstract: A recent result due to Teng Zhang compares the sum of $m$ matrices and the sum of their quadratic symmetric moduli: $$ \left| \sum_{k=1}^m A_k\right| \le \sqrt{2} \left| \sum_{k=1}^m |A_k|<em>{\qsym}\right| $$ for every unitarily invariant norm. Here $|A|</em>{\qsym}$ is the quadratic mean of $|A|$ and $|A<sup>*|$. We derive operator and eigenvalue refinements of Zhang's inequality from a new polar decomposition for the quadratic symmetric modulus. For instance, $$ \left| \sum_{k=1}^m A_k\right| \le \frac{\sqrt{2}}{2} \left{ \sum_{k=1}^m \left(|A_k|<em>{\qsym}+V|A_k|</em>{\qsym}V^{*\right)\right}} $$ for some unitary matrix $V$. We also establish the polar decomposition for the maximal modulus associated with Olson's order, and derive, as in the quadratic case, a series of estimates.