Cross-positive linear maps, positive polynomials and sums of squares

Klep, Igor (author), Šivic, Klemen (author), Zalar, Aljaž (author)

A $\ast$-linear map $\Phi$ between matrix spaces is cross-positive if it is positive on orthogonal pairs $(U,V)$ of positive semidefinite matrices in the sense that $\langle U,V \rangle:={\rm tr}(UV)=0$ implies $\langle\Phi (U),V \rangle \ge 0$, and is completely cross-positive if all its ampliations $I_n \otimes \Phi$ are cross-positive. (Completely) cross-positive maps arise in the theory of operator semigroups, where they are sometimes called exponentially-positive maps, and are also important in the theory of affine processes on symmetric cones in mathematical finance. To each $\Phi$ as above a bihomogeneous form is associated by $p_\Phi (x,y)=y^T\Phi (xx^T)y$. Then $\Phi$ is cross-positive if and only if $p_\Phi$ is nonnegative on the variety of pairs of orthogonal vectors $\{(x,y) | x^Ty = 0\}$. Moreover, $\Phi$ is shown to be completely cross-positive if and only if $p_\Phi$ is a sum of squares modulo the principal ideal $(x^Ty)$. These observations bring the study of cross-positive maps into the powerful setting of real algebraic geometry. Here this interplay is exploited to prove quantitative bounds on the fraction of cross-positive maps that are completely cross-positive. Detailed results about cross-positive maps $\Phi$ mapping between $3\times3$ matrices are given. Finally, an algorithm to produce cross-positive maps that are not completely cross-positive is presented.

• Language: English
• Keywords: positive polynomials, sum of squares, positive maps, completely positive maps, one-parameter semigroups, convex cones
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics; FRI - Faculty of Computer and Information Science
• Publication status: Published
• Publication version: Version of Record
• Publication date: 01.02.2026
• Year: 2026
• Number of pages: Str. 189-243
• Numbering: Vol. 688
• PID: 20.500.12556/RUL-175128
• UDC: 517.9
• ISSN on article: 0021-8693
• DOI: 10.1016/j.jalgebra.2025.09.018
• COBISS.SI-ID: 253591043
• Publication date in RUL: 17.10.2025

Record is a part of a journal

• Title: Journal of algebra
• Shortened title: J. algebra
• Publisher: Elsevier
• ISSN: 0021-8693
• COBISS.SI-ID: 1310986

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Secondary language

• Language: Slovenian
• Keywords: pozitivni polinomi, vsote kvadratov, pozitivne preslikave, popolnoma pozitivnaepreslikave, enoparametrične polgrupe, konveksni stožci