| Title: | Cross-positive linear maps, positive polynomials and sums of squares |
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| Authors: | ID Klep, Igor (Author)
ID Šivic, Klemen (Author)
ID Zalar, Aljaž (Author) |
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| Files: |  PDF - Presentation file, download (1,67 MB)
MD5: BC79CF7E265568629880C16626980E6A
 URL - Source URL, visit https://www.sciencedirect.com/science/article/pii/S0021869325005526
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| Language: | English |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
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| Abstract: | 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. |
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| Keywords: | positive polynomials, sum of squares, positive maps, completely positive maps, one-parameter semigroups, convex cones |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.02.2026 |
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| Year of publishing: | 2026 |
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| Number of pages: | str. 189-243 |
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| Numbering: | Vol. 688 |
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| PID: | 20.500.12556/DiRROS-23886  |
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| UDC: | 517.9 |
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| ISSN on article: | 0021-8693 |
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| DOI: | 10.1016/j.jalgebra.2025.09.018 |
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| COBISS.SI-ID: | 253591043 |
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| Publication date in DiRROS: | 17.10.2025 |
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| Views: | 180 |
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| Downloads: | 94 |
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