Korovkin-type results and doubly stochastic transformations over Euclidean Jordan algebras
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2022-09-27
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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.