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 $1\le p<\infty$. It was conjectured in [6] that a universal inequality $||{\sigma }_{\downarrow }(A)+{\sigma }_{\downarrow }(B)|{|}_p^p+||{\sigma }_{\downarrow }(A)-{\sigma }_{\downarrow }(B)|{|}_p^p\le ||A+B|{|}_p^p+||A-B|{|}_p^p\le ||{\sigma }_{\uparrow }(A)+{\sigma }_{\downarrow }(B)|{|}_p^p+||{\sigma }_{\uparrow }(A)-{\sigma }_{\downarrow }(B)|{|}_p^p$ might hold for $1\le p\le 2$ and reverse at $p\ge 2$, potentially providing a stronger inequality to the generalization of Hanner's Inequality to complex matrices $||A+B|{|}_p^p+||A-B|{|}_p^p\ge (||A|{|}_p+||B|{|}_p{)}^p+|||A|{|}_p-||B|{|}_p{|}^p$. 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 $p$.

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