On the finiteness of the cone spectrum of certain linear transformations on Euclidean Jordan algebras

Let L be a linear transformation on a finite dimensional real Hilbert space H and K be a closed convex cone with dual K ∗ in H . The cone spectrum of L relative to K is the set of all real λ for which the linear complementarity problem x ∈ K , y = L ( x ) - λ x ∈ K ∗ , and 〈 x , y 〉 = 0 admits a nonzero solution x . In the setting of a Euclidean Jordan algebra H and the corresponding symmetric cone K , we discuss the finiteness of the cone spectrum for Z -transformations and quadratic representations on H