Matrix rearrangement inequalities revisited

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