Some hybrid matrix triangle inequalities

A recent result due to Teng Zhang compares the sum of $m$ matrices and the sum of their quadratic symmetric moduli: $$\parallel \sum _{k=1}^m{A}_k\parallel \le \sqrt{2}\parallel \sum _{k=1}^m\mathrm{\mid }{A}_k{\mathrm{\mid }}_{\text{qsym}}\parallel$$ for every unitarily invariant norm. Here $\mathrm{\mid }A{\mathrm{\mid }}_{\text{qsym}}$ is the quadratic mean of $\mathrm{\mid }A\mathrm{\mid }$ and $\mathrm{\mid }{A}^{\ast }\mathrm{\mid }$. We derive operator and eigenvalue refinements of Zhang’s inequality from a new polar decomposition for the quadratic symmetric modulus. For instance, $$\mid \sum _{k=1}^m{A}_k\mid \le \frac{\sqrt{2}}{2}\left\{\sum _{k=1}^m(\mathrm{\mid }{A}_k{\mathrm{\mid }}_{\text{qsym}}+V\mathrm{\mid }{A}_k{\mathrm{\mid }}_{\text{qsym}}{V}^{\ast })\right\}$$ 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.