Matrix Rearrangement Inequalities Revisited

TL;DR

This paper investigates matrix rearrangement inequalities involving Schatten norms, providing counterexamples to universal inequalities, simplifying proofs via majorization, and extending Hanner's Inequality to specific matrix cases.

Contribution

It extends existing inequalities, offers counterexamples to conjectured universal inequalities, simplifies proofs using majorization, and generalizes Hanner's Inequality for certain matrix classes.

Findings

Counterexamples to universal rearrangement inequalities.

Simplified proofs using majorization techniques.

Extension of Hanner's Inequality to commuting and anticommuting matrices.

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 $p$.