A uniqueness result for linear complementarity problems over the Jordan spin algebra

Given a Euclidean Jordan algebra (V,∘,〈⋅,⋅〉) with the (corresponding) symmetric cone K, a linear transformation L:V→V and q∈V, the linear complementarity problem LCP(L,q) is to find a vector x∈V such thatx∈K,y:=L(x)+q∈Kandx∘y=0. To investigate the global uniqueness of solutions in the setting of Euclidean Jordan algebras, the P-property and its variants of a linear transformation were introduced in Gowda et al. (2004) [3] and it is shown that if LCP(L,q) has a unique solution for all q∈V, then L has the P-property but the converse is not true in general. In the present paper, when (V,∘,〈⋅,⋅〉) is the Jordan spin algebra, we show that LCP(L,q) has a unique solution for all q∈V if and only if L has the P-property and L is positive semidefinite on the boundary of K.