Korovkin-type results and doubly stochastic transformations over Euclidean Jordan algebras
| dc.contributor.author | Gowda, Muddappa | |
| dc.date.accessioned | 2022-10-21T16:07:54Z | |
| dc.date.available | 2022-10-21T16:07:54Z | |
| dc.date.issued | 2022-09-27 | |
| dc.description.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. | en_US |
| dc.description.sponsorship | Thanks are due to Michael Orlitzky, Roman 17 Sznajder, and Juyoung Jeong for their comments and suggestions. In a private communication [11], Jeong notes that Theorem 3.1 and Theorem 4.2 continue to hold when the quadratic function t 7→ t 2 on R is replaced by a strictly convex function. He also shows (by an example) that when n ≥ 4, the set Ωp (that appears in the problem posed in Section 3) may contain matrices other than doubly stochastic ones. | en_US |
| dc.description.uri | https://arxiv.org/abs/2209.13303 | en_US |
| dc.format.extent | 19 pages | en_US |
| dc.genre | journal articles | en_US |
| dc.genre | preprints | en_US |
| dc.identifier | doi:10.13016/m2ilja-bx27 | |
| dc.identifier.uri | https://doi.org/10.48550/arXiv.2209.13303 | |
| dc.identifier.uri | http://hdl.handle.net/11603/26219 | |
| dc.language.iso | en_US | en_US |
| dc.relation.isAvailableAt | The University of Maryland, Baltimore County (UMBC) | |
| dc.relation.ispartof | UMBC Mathematics Department Collection | |
| dc.relation.ispartof | UMBC Faculty Collection | |
| dc.rights | This item is likely protected under Title 17 of the U.S. Copyright Law. Unless on a Creative Commons license, for uses protected by Copyright Law, contact the copyright holder or the author. | en_US |
| dc.title | Korovkin-type results and doubly stochastic transformations over Euclidean Jordan algebras | en_US |
| dc.type | Text | en_US |