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

TL;DR

This paper extends Korovkin-type approximation results to Euclidean Jordan algebras, showing that positive linear transformations fixing certain elements are necessarily the identity on specific subspaces, with implications for doubly stochastic transformations.

Contribution

It introduces analogs of Korovkin's theorem within Euclidean Jordan algebras, characterizing positive linear transformations that fix particular elements and establishing conditions for them to be the identity.

Findings

Positive linear transformations fixing specific elements are the identity on the spectral subspace.

Such transformations are doubly stochastic if they fix a Jordan frame.

Sequential and weak-majorization versions of the results are also provided.

Abstract

A well-known theorem of Korovkin asserts that if $\{T_k\}$ is a sequence of positive linear transformations on $C[a,b]$ such that $T_k(h)\rightarrow h$ (in the sup-norm on $C[a,b]$) for all $h\in \{1,\phi,\phi^2\}$, where $\phi(t)=t$ on $[a,b]$, then $T_k(h)\rightarrow h$ for all $h\in C[a,b]$. In particular, if $T$ is a positive linear transformation on $C[a,b]$ such that $T(h)=h$ for all $h\in \{1,\phi,\phi^2\}$, 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 $h\in \{e,p,p^2\}$, 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.