POSITIVITY OF THE TRANSMUTATION OPERATORS AND ABSOLUTE CONTINUITY OF THEIR REPRESENTING MEASURES FOR A ROOT SYSTEM ON $\mathbb{R}^d$

Abstract

We consider the transmutation operators Vk,V W k and t Vk t V W k as- sociated respectively with the Cherednik operators and the Heckman-Opdam theory, called also in (9), (10) the trigonometric Dunkl intertwining operators and their dual paper. In this paper we prove that the operators Vk,V W k and t Vk, t V W k are positivity preserving and allows positive integral representations. Next we study the absolute continuity of their representing measures and we deduce that the Opdam-Cherednik kernel and the Heckman-Opdam hypergeo- metric function are positive definite.