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

Extensions of P-property, R0-property and semidefinite linear complementarity problems

In this manuscript, we present some new results for the semidefinite linear complementarity problem, in the context of three notions for linear transformations, viz., pseudo w-P property, pseudo Jordan w-P property and pseudo SSM property. Interconnections with the P#-property (proposed recently in the literature) is presented. We also study the R#-property of a linear transformation, extending the rather well known notion of an R0-matrix. In particular, results are presented for the Lyapunov, Stein, and the multiplicative transformation