Eventual cone invariance revisited
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866913232257548288 |
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| author | Glück, Jochen Hölz, Julian |
| author_facet | Glück, Jochen Hölz, Julian |
| contents | We consider finite-dimensional real vector spaces $X$ ordered by a closed cone $X_+$ with non-empty interior and study eventual nonnegativity of matrix semigroups $(e^{tA})_{t \ge 0}$ with respect to this cone.
Our first contribution is the observation that, for general cones, one needs to distinguish between different notions of eventual nonnegativity: (i) uniform eventual nonnegativity means that $e^{tA}$ maps $X_+$ into $X_+$ for all sufficiently large times $t$; (ii) individual eventual nonnegativity means that for each $x \in X_+$ the vector $e^{tA}x$ is in $X_+$ for all $t$ larger than an $x$-dependent time $t_0$; and (iii) weak eventual nonnegativity means that for each $x \in X_+$ and each functional $x'$ in the dual cone $X'_+$ the value $\langle x', e^{tA} x \rangle$ is in $[0,\infty)$ for all $t$ larger than an $x$- and $x'$-dependent time $t_0$. Until now, only the first of these notions has been studied in the literature. We demonstrate by examples that, somewhat surprisingly for finite-dimensional spaces, all three notions are different.
Our second contribution is to show that typical Perron-Frobenius like properties remain valid under the weakest of the above notions.
Third, we study a strengthened form of the above mentioned concepts, namely eventual positivity. We prove that uniform, individual and weak versions of this property are - in contrast to the nonnegative case - equivalent, and that they can be characterized by spectral properties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_07809 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Eventual cone invariance revisited Glück, Jochen Hölz, Julian Rings and Algebras Functional Analysis 15B48, 15A16, 15A18, 46B40, 47B65 We consider finite-dimensional real vector spaces $X$ ordered by a closed cone $X_+$ with non-empty interior and study eventual nonnegativity of matrix semigroups $(e^{tA})_{t \ge 0}$ with respect to this cone. Our first contribution is the observation that, for general cones, one needs to distinguish between different notions of eventual nonnegativity: (i) uniform eventual nonnegativity means that $e^{tA}$ maps $X_+$ into $X_+$ for all sufficiently large times $t$; (ii) individual eventual nonnegativity means that for each $x \in X_+$ the vector $e^{tA}x$ is in $X_+$ for all $t$ larger than an $x$-dependent time $t_0$; and (iii) weak eventual nonnegativity means that for each $x \in X_+$ and each functional $x'$ in the dual cone $X'_+$ the value $\langle x', e^{tA} x \rangle$ is in $[0,\infty)$ for all $t$ larger than an $x$- and $x'$-dependent time $t_0$. Until now, only the first of these notions has been studied in the literature. We demonstrate by examples that, somewhat surprisingly for finite-dimensional spaces, all three notions are different. Our second contribution is to show that typical Perron-Frobenius like properties remain valid under the weakest of the above notions. Third, we study a strengthened form of the above mentioned concepts, namely eventual positivity. We prove that uniform, individual and weak versions of this property are - in contrast to the nonnegative case - equivalent, and that they can be characterized by spectral properties. |
| title | Eventual cone invariance revisited |
| topic | Rings and Algebras Functional Analysis 15B48, 15A16, 15A18, 46B40, 47B65 |
| url | https://arxiv.org/abs/2303.07809 |