Generalizing Lee's conjecture on the sum of absolute values of matrices

Abstract

Let $\Vert \cdot {\Vert }_p$ denote the Schatten $p$-norm of matrices and $\Vert \cdot {\Vert }_F$ the Frobenius norm. For a square matrix $X$, let $|X|$ denote its absolute value. In 2010, Eun-Young Lee posed the problem of determining the smallest constant $c_p$ such that $\Vert A+B{\Vert }_p\le c_p\Vert |A|+|B|{\Vert }_p$ for all complex matrices $A,B$. The Frobenius case $(p=2)$ conjectured by Lee was proved by Lin and Zhang (2022)~\cite{LinZhang2022} and re-proved by Zhang (2025)~\cite{Zhang2025}. In this paper, we extend Lee's conjecture from two matrices to an arbitrary number $m\ge 2$ of complex matrices $A_1,\dots ,A_m$, and determine the sharp inequality $${\Vert \sum _{k=1}^m{A}_k\Vert }_F\le \sqrt{\frac{1+\sqrt{m}}{2}}{\Vert \sum _{k=1}^m|A_k|\Vert }_F,$$ with equality attained by an equiangular rank-one family. We further generalize Lee's problem by seeking the smallest constant $c_p(m)$ such that $\Vert {\sum }_{k=1}^m{A}_k{\Vert }_p\le c_p(m)\Vert {\sum }_{k=1}^m|A_k|{\Vert }_p$. It is shown that $c_p(m)\le (\sqrt{m}{)}^{1-1/p}$, and we conjecture a closed-form expression for the optimal value of $c_p(m)$ that recovers all known cases $p=1,2,\infty$.

7 pages. This is the version accepted for publication in Linear Algebra and Its Applications