Cross-positive linear maps, positive polynomials and sums of squares
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| Format: | Preprint |
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2024
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| _version_ | 1866909899872534528 |
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| author | Klep, Igor Šivic, Klemen Zalar, Aljaž |
| author_facet | Klep, Igor Šivic, Klemen Zalar, Aljaž |
| contents | A linear map $Φ$ between matrix spaces is called cross-positive if it is positive on orthogonal pairs $(U,V)$ of positive semidefinite matrices in the sense that $\langle U,V\rangle:=\text{Tr}(UV)=0$ implies $\langle Φ(U),V\rangle\geq0$, and is completely cross-positive if all its ampliations $I_n\otimes Φ$ 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 $Φ$ as above a bihomogeneous form is associated by $p_Φ(x,y)=y^TΦ(xx^T)y$. Then $Φ$ is cross-positive if and only if $p_Φ$ is nonnegative on the variety of pairs of orthogonal vectors $\{(x,y)\mid x^Ty=0\}$. Moreover, $Φ$ is shown to be completely cross-positive if and only if $p_Φ$ 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 $Φ$ mapping between $3\times 3$ matrices are given. Finally, an algorithm to produce cross-positive maps that are not completely cross-positive is presented. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_17425 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cross-positive linear maps, positive polynomials and sums of squares Klep, Igor Šivic, Klemen Zalar, Aljaž Functional Analysis 13J30, 15B48, 46L07, 47D06, 52A40 (Primary), 47L25, 90C22 (Secondary) A linear map $Φ$ between matrix spaces is called cross-positive if it is positive on orthogonal pairs $(U,V)$ of positive semidefinite matrices in the sense that $\langle U,V\rangle:=\text{Tr}(UV)=0$ implies $\langle Φ(U),V\rangle\geq0$, and is completely cross-positive if all its ampliations $I_n\otimes Φ$ 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 $Φ$ as above a bihomogeneous form is associated by $p_Φ(x,y)=y^TΦ(xx^T)y$. Then $Φ$ is cross-positive if and only if $p_Φ$ is nonnegative on the variety of pairs of orthogonal vectors $\{(x,y)\mid x^Ty=0\}$. Moreover, $Φ$ is shown to be completely cross-positive if and only if $p_Φ$ 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 $Φ$ mapping between $3\times 3$ matrices are given. Finally, an algorithm to produce cross-positive maps that are not completely cross-positive is presented. |
| title | Cross-positive linear maps, positive polynomials and sums of squares |
| topic | Functional Analysis 13J30, 15B48, 46L07, 47D06, 52A40 (Primary), 47L25, 90C22 (Secondary) |
| url | https://arxiv.org/abs/2401.17425 |