On (conditional) positive semidefiniteness in a matrix-valued context

Fritz Gesztesy¹ ; Michael Pang²

¹ Department of Mathematics University of Missouri Columbia, MO 65211, U.S.A. and Department of Mathematics Baylor University One Bear Place #97328 Waco, TX 76798-7328, U.S.A.
² Department of Mathematics University of Missouri Columbia, MO 65211, U.S.A.

Studia Mathematica, Tome 236 (2017) no. 2, pp. 143-192

Résumé

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.

Détail

DOI : 10.4064/sm8531-7-2016

Keywords: nutshell intend extend schoenberg classical theorem connecting conditionally positive semidefinite functions mathbb mathbb mathbb their positive semidefinite exponentials exp matrix valued functions colon mathbb mathbb times mathbb moreover study closely associated property exp nabla positivity preserving its failure extend directly matrix valued context