Generalizing Lee's conjecture on the sum of absolute values of matrices
Quanyu Tang, Shu Zhang
TL;DR
This work addresses generalizing Lee's conjecture on the sum of absolute values of matrices to a finite family of $m$ matrices under Schatten norms. It develops a sharp Frobenius inequality via polar decompositions, contraction techniques, and weak majorisation, establishing $\left\|\sum_{k=1}^m A_k\right\|_F \le \sqrt{\frac{1+\sqrt m}{2}}\;\left\|\sum_{k=1}^m |A_k|\right\|_F$ with equality for equiangular rank-one systems, and shows the result extends to Hilbert–Schmidt operators and von Neumann algebras. The work then generalizes to arbitrary Schatten $p$-norms, proving a universal bound $c_p(m)\le (\sqrt m)^{1-1/p}$ and proposing a closed-form conjecture for the optimal $c_p(m)$ in terms of $x_{p,m}>0$ solving $x^p-2x-(m-1)=0$, which recovers the known cases $p=1,2,\infty$. These findings broaden Lee's problem from two matrices to arbitrary finite families and connect sharp constants to equiangular constructions, with potential implications for operator inequalities.
Abstract
Let $\|\!\cdot\!\|_p$ denote the Schatten $p$-norm of matrices and $\|\!\cdot\!\|_F$ 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 $\|A+B\|_p \le c_p\|\,|A|+|B|\,\|_p$ 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 $$ \left\|\sum_{k=1}^{m} A_k\right\|_F \le \sqrt{\frac{1+\sqrt{m}}{2}}\; \left\|\sum_{k=1}^{m}|A_k|\right\|_F , $$ 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 $ \|\sum_{k=1}^{m} A_k\|_p \le c_p(m)\, \|\sum_{k=1}^{m}|A_k|\|_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 $p=1,2,\infty$.