The positivity of the transmutation operators associated to the Cherednik operators for the root system $BC_2$

We consider the transmutation operators V_k, ^tV_k and V ^W _k , ^tV ^W _k associated respectively with the Cherednik operators and the Heckman-Opdam theory attached to the root system BC2, called also in [8, 9, 10] the trigonometric Dunkl intertwining operators, and their dual. In this paper we prove that the operators V_k, ^tV_k and V^W_k , ^tV^W_k are positivity preserving and allows positive integral representations. In particular we deduce that the Opdam-Cherednik and the Heckman-Opdam kernels are positive definite.