On the finiteness of the cone spectrum of certain linear transformations on Euclidean Jordan algebras (Q1030740)

Let \(H\) be a finite-dimensional real Hilbert space, \(K\) a closed convex cone in \(H\), \(K^\star\) its dual and \(L: H\longrightarrow H\) a linear map. The \textit{cone spectrum of} \(L\) relative to \(K\) is the set of all the real numbers \(\lambda \) for which there exists a nonzero element \(x\) in \(K\) with \(y= L(x)-\lambda x \in K^\star\) satisfying \((x, y) =0\). An \textit{Euclidean Jordan algebra} \(V\) is a Jordan algebra with a unit element whose underlying vector space is a finite-dimensional real Hilbert space with inner product \(( . , . )\) satisfying \((xy, z) = (x, yz)\) for every \(x, y, z\in V\). The closed convex cone of all the squares in \(V\) is a symmetric cone which is called the \textit{closed symmetric cone} of \(V\). As in every Jordan algebra, for all \(a\in V\) are denoted by \(L_a\) and \(P_a\) the linear operators defined by \(L_a (x) = ax\) and \(P_a (x) = 2a(ax)-a^2 x.\) In the paper under review it is explicitily computed the cone spectrum of the multiplication operators \(L_a\) and of the quadratic operators \(P_a\) of an Euclidean Jordan algebra relative to its closed symmetric cone. Furthermore, the authors prove a finite cone spectrum property for a special class of cones (proper cones) and of linear maps (\(\mathbb{Z}\)-transformations).