Schatten范数的严格拟逆Minkowski不等式
Sharp Quasi-Reverse Minkowski Inequality for Schatten Norms
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中文总结 AI 辅助
该论文针对Schatten p-范数，证明2≤p<∞时存在严格拟逆Minkowski不等式，对1<p<2则说明最优常数公式不成立并给出反例与解析构造。
中文摘要 AI 辅助
设‖·‖ₚ表示Schatten p-范数，|A|=(A*A)^(1/2)。对2≤p<∞，令xₚ>1为xₚᵖ=2xₚ+1的唯一解，定义Cₚ=√[xₚ(xₚ+1)]/(xₚᵖ+1)^(1/p)。我们证明了任意大小的复矩阵均满足严格不等式‖A+B‖ₚ≤Cₚ‖|A|+|B|‖ₚ，等价于若q=p/(p-1)且R,X,Y为半正定矩阵，则‖RX‖₁+‖RY‖₁≤Cₚ‖R‖ᵠ‖X+Y‖ₚ。对1<p<2，我们还证明了最优常数公式不成立，并给出数值反例与系统解析构造。
英文摘要
Let $\|\cdot\|_p$ denote the Schatten $p$-norm and let $|A|=(A^*A)^{1/2}$. For $2\leq p<\infty$, let $x_{p,m}>1$ be the unique solution of $x_{p,m}^p=2x_{p,m}+m-1$, and set \[ C_{p,m}=\frac{\sqrt{x_{p,m}(x_{p,m}+m-1)}}{(x_{p,m}^p+m-1)^{1/p}}. \] We prove the sharp inequality \[ \|A_1+\cdots+A_m\|_p\leq C_{p,m}\bigl\||A_1|+\cdots+|A_m|\bigr\|_p \] for arbitrary complex matrices of arbitrary size. Equivalently, if $q=p/(p-1)$ and $R,X_1,\cdots,+X_m$ are positive semidefinite, then \[ \|RX_1\|_1+\cdots\|RX_m\|_1 \leq C_{p,m}\|R\|_q\|X_1+\cdots X_m\|_p. \] For $1<p<2$, we also show that the formula proposed for the optimal constant fails. We give both a numerical counterexample and a systematic analytic construction.oposed for the optimal constant fails. We give both a numerical counterexample and a systematic analytic construction.
