逐元素正性保持算子的有限阶刻画

中文摘要 AI 辅助

设$I=(0,\rho)$，其中$0<\rho\leq\infty$，记$\mathbb{P}_n(I)$为所有元素属于$I$的实$n\times n$半正定矩阵构成的锥。矩阵理论中一个长期存在的问题是，刻画满足逐元素演算$f[A]=(f(a_{ij}))$对所有$A=(a_{ij})\in\mathbb{P}_n(I)$都保持半正定性的函数$f:I\to\mathbb{R}$。我们通过与$f$关联的欧拉-汉克尔矩阵，给出了这类保持算子的显式函数论刻画。我们首先对属于$C^{2n-2}(I)$的函数$f$给出刻画：记$\mathcal{E}=x\frac{d}{dx}$，欧拉-汉克尔矩阵为$\mathcal{H}_n(f;x)=\bigl[\mathcal{E}^{i+j}f(x)\bigr]_{i,j=0}^{n-1}$，我们证明$f[-]$在$\mathbb{P}_n(I)$上保持正性当且仅当对$0\leq k\leq n-1$有$f^{(k)}(x)\geq0$，且对每个$x\in I$有$\mathcal{H}_n(f;x)\succeq0$。证明结合了Karlin关于加性汉克尔核的有限阶准则，以及Khare和Tao的扩展原理。随后我们说明如何完全去除光滑性假设，对一般函数得到相同刻画，此时上述条件被解释为分布间的不等式。最后我们证明，我们的结果可还原二维情形下Vasudeva的光滑刻画、幂函数的FitzGerald-Horn临界指数，以及Belton-Guillot-Khare-Putinar对$\mathbb{P}_n(I)$上次数不超过$n$的多项式保持算子的刻画；还将Khare-Tao关于保持秩1半正定矩阵的实幂和的刻画，扩展到保持整个锥$\mathbb{P}_n(I)$的实幂和的情形。

英文摘要

Fix $I = (0,ρ)$, where $0<ρ\leq\infty$, and let $\mathbb{P}_n(I)$ be the set of positive semidefinite $n\times n$ matrices with entries in $I$. A longstanding problem in matrix theory is to characterize the functions $f: I \to \mathbb{R}$ for which the entrywise calculus $f[A] = [f(a_{ij})]_{i,j = 1}^{n}$ preserves positive semidefiniteness for all $A \in \mathbb{P}_n(I)$. We characterize these functions exactly: if $f\in C^{2n-2}(I)$ and $\mathcal{E} = x\frac{d}{dx}$, then this holds if and only if $$ f^{(k)}(x)\geq 0 \quad (0\leq k\leq n-1) \qquad\text{and}\qquad \bigl[\mathcal{E}^{i+j}f(x)\bigr]_{i,j = 0}^{n-1}\succeq 0 $$ for every $x\in I$. Regularization then removes all a priori smoothness: for $n\geq2$, every preserver belongs to $C^{2n-4}(I)$ and the same characterization holds by interpreting the last two derivatives in the sense of distributions. As applications, we recover classical results of FitzGerald--Horn and Vasudeva, and obtain a complete classification of generalized polynomials with prescribed real exponents and arbitrary coefficients. We also determine optimal constants in entrywise domination inequalities under finite regularity, extend the sharp finite-sum thresholds of Belton--Guillot--Khare--Putinar and Khare--Tao to positive mixtures of powers, and answer a question of Khare and Tao by showing that no finite collection of matrices with entries strictly inside $I$ can detect positivity preservation on $\mathbb{P}_n(I)$.

发表机构

• Université Laval（拉瓦尔大学）
• University of Delaware（特拉华大学）

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