Averages over matrix unitary orbits and spectral order

We establish matrix versions of the comparisons between the ${\mathrm{\ell }}^p$-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 $$\mid \sum _{k=1}^m{A}_k\mid \le \frac{1}{d}\sum _{i=1}^d{V}_i{\left\{\sum _{k=1}^m\mathrm{\mid }{A}_k{\mathrm{\mid }}^q\right\}}^{1\mathrm{/}q}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}{V}_i^{\ast }$$ 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.