2025/10/19 by Q.W Tang, Tang, Quanyu, Zhang Shu +1
Computer Science · Engineering · Mathematics · #Matrix Theory and Algorithms #graph theory and CDMA systems #Mathematical Inequalities and Applications
paper · pdf · doi:10.48550/arxiv.2510.16846
Let ‖ ⋅ ‖p denote the Schatten p-norm of matrices and ‖ ⋅ ‖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 cp such that ‖A+B‖p ≤ cp‖ |A|+|B| ‖p for all complex matrices A,B. The Frobenius case (p=2) conjectured by Lee was proved by Lin and Zhang (2022)~\citeLinZhang2022 and re-proved by Zhang (2025)~\citeZhang2025. In this paper, we extend Lee's conjecture from two matrices to an arbitrary number m ≥ 2 of complex matrices A1,…,Am, and determine the sharp inequality ‖∑k=1m Ak‖F ≤ √((1+√(m))/(2)) ‖∑k=1m|Ak|‖F , with equality attained by an equiangular rank-one family. We further generalize Lee's problem by seeking the smallest constant cp(m) such that ‖∑k=1m Ak‖p ≤ cp(m) ‖∑k=1m|Ak|‖p . It is shown that cp(m)≤ (√(m))1-1/p, and we conjecture a closed-form expression for the optimal value of cp(m) that recovers all known cases p=1,2,∞.