A Finite-order Characterization of Entrywise Positivity Preservers

Abstract

Fix $I=(0,\rho )$, where $0<\rho \le \infty$, 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 $$f^{(k)}(x)\ge 0(0\le k\le n-1)\text{and}[{\mathcal{E}}^{i+j}f(x){]}_{i,j=0}^{n-1}⪰0$$ for every $x\in I$. Regularization then removes all a priori smoothness: for $n\ge 2$, 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 ${\mathbb{P}}_n(I)$.