Korovkin-type results and doubly stochastic transformations over Euclidean Jordan algebras

Abstract

A well-known theorem of Korovkin asserts that if {Tk} is a sequence of positive linear transformations on C[a, b] such that Tk(h) → h (in the sup-norm on C[a, b]) for all h ∈ {1, φ, φ2}, where φ(t) = t on [a, b], then Tk(h) → h for all h ∈ C[a, b]. In particular, if T is a positive linear transformation on C[a, b] such that T (h) = h for all h ∈ {1, φ, φ2}, then T is the Identity transformation. In this paper, we present some analogs of these results over Euclidean Jordan algebras. We show that if T is a positive linear transformation on a Euclidean Jordan algebra V such that T (h) = h for all h ∈ {e, p, p2}, where e is the unit element in V and p is an element of V with distinct eigenvalues, then T = T ∗ = I (the Identity transformation) on the span of the Jordan frame corresponding to the spectral decomposition of p; consequently, if a positive linear transformation coincides with the Identity transformation on a Jordan frame, then it is doubly stochastic. We also present sequential and weak-majorization versions.