On (conditional) positive semidefiniteness in a matrix-valued context
Volume 236 / 2017
Studia Mathematica 236 (2017), 143-192
MSC: Primary 42A82, 42B15, 43A35; Secondary 43A15, 46E40, 46G10.
DOI: 10.4064/sm8531-7-2016
Published online: 2 December 2016
Abstract
In a nutshell, we intend to extend Schoenberg’s classical theorem connecting conditionally positive semidefinite functions $F : \mathbb{R}^n \to \mathbb{C}$, $n \in \mathbb{N}$, and their positive semidefinite exponentials $\exp(tF)$, $t \gt 0$, to the case of matrix-valued functions $F \colon \mathbb{R}^n \to \mathbb{C}^{m \times m}$, $m \in \mathbb{N}$. Moreover, we study the closely associated property that $\exp(t F(- i \nabla))$, $t \gt 0$, is positivity preserving and its failure to extend directly in the matrix-valued context.
