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

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