Analog of Schoenberg’s Theorem for $a$-Conditionally Negative Definite Matrix-Valued Kernels

Abstract

Schoenberg’s classical 1938 theorem asserts that, given a function $\rho\colon G\times G\to\mathbb{C}$, the function $\exp(-t\rho)$ is a positive definite kernel on $G\times G$ for any $t>0$ if and only if the kernel $\rho$ is Hermitian and negative definite on $G\times G$. An analog of this theorem for matrices was essentially proved by C. Löwner in 1966. Recently (in 2021), C. Dörr and M. Schlather obtained an analog of Schoenberg’s theorem for real matrix-valued functions $\rho(x)$, $x\in \mathbb{R}^d$. This analog refers to conditionally negative definite matrix-valued functions. In the present paper, $a$-conditionally negative definite matrix-valued kernels $\rho$ on $G\times G$ for which an analog of Schoenberg’s theorem holds are introduced and studied. The following more general problem is also considered: for what functions $f$ and $g$ and matrix-valued kernels $\rho$ on $G\times G$ is the function $f(tg(\rho))$ a positive definite matrix-valued kernel on $G\times G$ for any $t>0$? Necessary conditions, sufficient conditions, and examples of such functions are given.