论文标题
重新审视矩阵重排的不平等现象
Matrix Rearrangement Inequalities Revisited
论文作者
论文摘要
令$ || x || _p = \ text {tr} [（x^\ ast x）^{p/2}]^{1/p} $表示m_ {n \ times n}（n \ times n}（\ mathbb {c}）$ x \ in M_ { $ \ downarrow $表示其增加或减少重排。我们希望研究$ || a+b || _p^p+|| a-a-a-b || _p^p $，$ || ||σ_\ downarrow（a）+σ_\ downarrow（b）|| _p^p+|| _p^p+| | || = ||σ_ $ ||σ_\ uparrow（a）+σ_\ downarrow（b）|| _p^p+|| = ||σ_\ uparrow（a） - σ_\ downarrow（b）|| _p^p $ for $ 1 \ leq p <\ p <\ infty $。在[6]中指出，通用不平等$ ||σ_\ downarrow（a）+σ_\ downarrow（b）|| _p^p+|| _p+||σ_\ downarrow（a）-townarrow（a）-townarrow（a）-townarrow（a）-t \ downarrow（b） ||σ_\uparrow(A)+σ_\downarrow(B)||_p^p+||σ_\uparrow(A)-σ_\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矩阵$ || a+b || _p^p+|| a-a-b || _p^p \ geq（|| a || a || _p+|| b || _p）^p+|||我们扩展了[5]的不平等现象的一些案例，但为任何一般的重排不平等持有提供了反例。我们使用多数化技术简化了[6]的原始证明。这也使我们能够表征所考虑的所有不平等现象的平等案例。我们还直接解决了通勤，统一和$ \ {a，b \} = 0 $案例，并扩展抗议机的作用。为此，我们将Hanner的自我拥护矩阵不平等扩展到$ \ {a，b \} = 0 $ case for $ p $的所有范围。
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 $σ(X)$ the singular values with $\uparrow$ $\downarrow$ indicating its increasing or decreasing rearrangements. We wish to examine inequalities between $||A+B||_p^p+||A-B||_p^p$, $||σ_\downarrow(A)+σ_\downarrow(B)||_p^p+||σ_\downarrow(A)-σ_\downarrow(B)||_p^p$, and $||σ_\uparrow(A)+σ_\downarrow(B)||_p^p+||σ_\uparrow(A)-σ_\downarrow(B)||_p^p$ for various values of $1\leq p<\infty$. It was conjectured in [6] that a universal inequality $||σ_\downarrow(A)+σ_\downarrow(B)||_p^p+||σ_\downarrow(A)-σ_\downarrow(B)||_p^p\leq ||A+B||_p^p+||A-B||_p^p \leq ||σ_\uparrow(A)+σ_\downarrow(B)||_p^p+||σ_\uparrow(A)-σ_\downarrow(B)||_p^p$ 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 $p$.
