On the nonnegative rank of positive operators
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910241033027584 |
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| author | Drnovšek, Roman Kandić, Marko |
| author_facet | Drnovšek, Roman Kandić, Marko |
| contents | In this paper we introduce the concept of a nonnegative rank of a positive operator $T\colon X\to Y$ between ordered vector spaces. In the case of nonnegative matrices, our definition agrees with the standard definition of a nonnegative rank. Under some natural and mild assumptions on the cone $Y_+$, we prove that the nonnegative rank and the rank agree whenever the rank is at most two. This can be considered as the infinite-dimensional version of \cite[Theorem 4.1]{CR93}. We also provide an example of a positive rank-three operator on the Banach lattice $C[0,1]$ with an infinite nonnegative rank.exceed $\lceil 6\min\{m,n\}/7\rceil$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_21232 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the nonnegative rank of positive operators Drnovšek, Roman Kandić, Marko Functional Analysis 47B65, 46A40 In this paper we introduce the concept of a nonnegative rank of a positive operator $T\colon X\to Y$ between ordered vector spaces. In the case of nonnegative matrices, our definition agrees with the standard definition of a nonnegative rank. Under some natural and mild assumptions on the cone $Y_+$, we prove that the nonnegative rank and the rank agree whenever the rank is at most two. This can be considered as the infinite-dimensional version of \cite[Theorem 4.1]{CR93}. We also provide an example of a positive rank-three operator on the Banach lattice $C[0,1]$ with an infinite nonnegative rank.exceed $\lceil 6\min\{m,n\}/7\rceil$. |
| title | On the nonnegative rank of positive operators |
| topic | Functional Analysis 47B65, 46A40 |
| url | https://arxiv.org/abs/2605.21232 |