A finite-order characterization of entrywise positivity preservers

Abstract

Let $I=(0,ρ)$, where $0<ρ\leq\infty$, and let $\mathbb{P}_n(I)$ denote the cone of real positive semidefinite $n\times n$ matrices whose entries belong to $I$. A longstanding problem in matrix theory is to characterize the functions $f: I \to \mathbb{R}$ for which the entrywise calculus $f[A] = (f(a_{ij}))$ preserves positive semidefiniteness for all $A = (a_{ij}) \in \mathbb{P}_n(I)$. We provide an explicit function-theoretic characterization of such preservers via the Euler--Hankel matrix associated to $f$. We begin by providing the characterization for functions $f \in C^{2n-2}(I)$. Writing $\mathcal{E} =x\frac{d}{dx}$ and denoting the Euler--Hankel matrix by $$ \mathcal{H}_n(f;x)=\bigl[\mathcal{E}^{i+j}f(x)\bigr]_{i,j=0}^{n-1}, $$ we prove that $f[-]$ preserves positivity on $\mathbb{P}_n(I)$ if and only if $f^{(k)}(x) \geq0$ for $0\leq k\leq n-1$ and $\mathcal{H}_n(f;x)\succeq 0$ for every $x\in I$. The proof combines Karlin&#39;s finite-order criterion for additive Hankel kernels with an extension principle of Khare and Tao. We then show how the smoothness hypothesis can be entirely removed to obtain the same characterization for general functions, where the above conditions are interpreted as inequalities between distributions. We conclude by showing how our results recover the smooth form of Vasudeva&#39;s characterization in dimension two, the FitzGerald--Horn critical exponent for power functions, and the characterization of Belton--Guillot--Khare--Putinar of polynomial preservers of degree at most $n$ on $\mathbb{P}_n(I)$. Finally, we extend the Khare--Tao characterization of sums of real powers preserving rank $1$ positive semidefinite matrices to sums of real powers that preserve the full cone $\mathbb{P}_n(I)$.