Invertible positive maps that are not automorphism
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914567329677312 |
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| author | Raickwade, Pavankumar Sivakumar, K. C. |
| author_facet | Raickwade, Pavankumar Sivakumar, K. C. |
| contents | Let $X$ be a real normed vector space with a cone $K\subseteq X$ satisfying either (i) $K$ is closed with non-empty interior or (ii) $K$ has non-zero extremals or (iii) $K$ is closed and $X$ is a Banach space. In this short note, we provide a method to construct an invertible linear map $T\colon X\to X$ such that $T[K]\subseteq K$ but $T^{-1}[K]\not\subseteq~K$. In particular, we show that, for every cone automorphism $S\colon X\to X$, there exists a rank one perturbation of $S$ which is positive and invertible, but does not have a positive inverse. We provide examples from four diverse situations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_14739 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Invertible positive maps that are not automorphism Raickwade, Pavankumar Sivakumar, K. C. Functional Analysis 47B60, 47B65 Let $X$ be a real normed vector space with a cone $K\subseteq X$ satisfying either (i) $K$ is closed with non-empty interior or (ii) $K$ has non-zero extremals or (iii) $K$ is closed and $X$ is a Banach space. In this short note, we provide a method to construct an invertible linear map $T\colon X\to X$ such that $T[K]\subseteq K$ but $T^{-1}[K]\not\subseteq~K$. In particular, we show that, for every cone automorphism $S\colon X\to X$, there exists a rank one perturbation of $S$ which is positive and invertible, but does not have a positive inverse. We provide examples from four diverse situations. |
| title | Invertible positive maps that are not automorphism |
| topic | Functional Analysis 47B60, 47B65 |
| url | https://arxiv.org/abs/2605.14739 |