Averages over matrix unitary orbits and spectral order

Abstract

We establish matrix versions of the comparisons between the ${\ell }^p$-norms or quasi-norms for sequences of complex numbers. For instance, given $1\ge q>0$, and a family of $m$ normal $d\times d$ matrices $A_1,\dots ,A_m$, we show that $$\left|\sum _{k=1}^m{A}_k\right|\le \frac{1}{d}\sum _{i=1}^d{V}_i{\left\{\sum _{k=1}^m|A_k{|}^q\right\}}^{1/q}{V}_i^{\ast }$$ for some unitary $d\times d$ matrices $V_1,\dots ,V_d$. We also give applications to Olson's spectral order and to the comparison between the symmetric modulus and the quadratic symmetric modulus. In particular we show that the sum $A+B$ of two positive matrices submajorizes their Kato supremum $A\vee B$, thereby completing majorization results due to Ando.

Corrected version; 17 pages