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 $A_1,\ldots, A_m$, we show that $$ \left|\sum_{k=1}^m A_k\right| \le \frac{1}{d}\sum_{i=1}^d V_i\left\{\sum_{k=1}^m |A_k|^{q}\right\}^{1/q}\!\!\!\!V_i^* $$ for some unitary $d\times d$ matrices $V_1,\ldots, V_d$. 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.
From: Jean-Christophe Bourin [view email]
[v1]
Sun, 14 Jun 2026 06:21:15 UTC (13 KB)
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