Matrix Rearrangement Inequalities Revisited

Abstract

Let $||X||_p=\text{Tr}[(X^\ast X)^{p/2}]^{1/p}$ denote the $p$-Schatten norm of a matrix $X\in M_{n\times n}(\mathbb{C})$, and $\sigma(X)$ the singular values with $\uparrow$ $\downarrow$ indicating its increasing or decreasing rearrangements. We wish to examine inequalities between $||A+B||_p^p+||A-B||_p^p$, $||\sigma_\downarrow(A)+\sigma_\downarrow(B)||_p^p+||\sigma_\downarrow(A)-\sigma_\downarrow(B)||_p^p$, and $||\sigma_\uparrow(A)+\sigma_\downarrow(B)||_p^p+||\sigma_\uparrow(A)-\sigma_\downarrow(B)||_p^p$ for various values of $1\leq p<\infty$. It was conjectured in [6] that a universal inequality $||\sigma_\downarrow(A)+\sigma_\downarrow(B)||_p^p+||\sigma_\downarrow(A)-\sigma_\downarrow(B)||_p^p\leq ||A+B||_p^p+||A-B||_p^p \leq ||\sigma_\uparrow(A)+\sigma_\downarrow(B)||_p^p+||\sigma_\uparrow(A)-\sigma_\downarrow(B)||_p^p$ might hold for $1\leq p\leq 2$ and reverse at $p\geq 2$, potentially providing a stronger inequality to the generalization of Hanner's Inequality to complex matrices $||A+B||_p^p+||A-B||_p^p\geq (||A||_p+||B||_p)^p+|||A||_p-||B||_p|^p$. We extend some of the cases in which the inequalities of [5] hold, but offer counterexamples to any general rearrangement inequality holding. We simplify the original proofs of [6] with the technique of majorization. This also allows us to characterize the equality cases of all of the inequalities considered. We also address the commuting, unitary, and $\{A,B\}=0$ cases directly, and expand on the role of the anticommutator. In doing so, we extend Hanner's Inequality for self-adjoint matrices to the $\{A,B\}=0$ case for all ranges of $p$.