Product of nonnegative selfadjoint operators in unbounded settings
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908456717385728 |
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| author | Barkaoui, Yosra Hassi, Seppo |
| author_facet | Barkaoui, Yosra Hassi, Seppo |
| contents | In this paper, necessary and sufficient conditions are established for the factorization of a closed, in general, unbounded operator $T=AB$ into a product of two nonnegative selfadjoint operators $A$ and $B.$ Already the special case, where $A$ or $B$ is bounded, leads to new results and is of wider interest, since the problem is connected to the notion of similarity of the operator $T$ to a selfadjoint one, but, in fact, goes beyond this case. It is proved that this subclass of operators can be characterized not only by means of quasi-affinity of $T^*$ to an operator $S=S^* \geq 0$, but also via Sebestyén inequality, a result known in the setting of bounded operators $T.$ Another subclass of operators $T,$ where $A$ or $B$ has a bounded inverse, leads to a similar analysis. This gives rise to a reversed version of Sebestyén inequality which is introduced in the present paper. It is shown that this second subclass, where $A^{-1}$ or $B^{-1}$ is bounded, can be characterized in a similar way by means of quasi-affinity of $T,$ rather that $T^*,$ to an operator $S=S^*\geq 0$. Furthermore, the connection between these two classes and weak-similarity as well as quasi-similarity to some $S=S^*\geq 0$ is investigated. Finally, the special case where $ S$ is bounded is considered. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_14404 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Product of nonnegative selfadjoint operators in unbounded settings Barkaoui, Yosra Hassi, Seppo Functional Analysis 47A62, 47B02, 47B25, 47A06 In this paper, necessary and sufficient conditions are established for the factorization of a closed, in general, unbounded operator $T=AB$ into a product of two nonnegative selfadjoint operators $A$ and $B.$ Already the special case, where $A$ or $B$ is bounded, leads to new results and is of wider interest, since the problem is connected to the notion of similarity of the operator $T$ to a selfadjoint one, but, in fact, goes beyond this case. It is proved that this subclass of operators can be characterized not only by means of quasi-affinity of $T^*$ to an operator $S=S^* \geq 0$, but also via Sebestyén inequality, a result known in the setting of bounded operators $T.$ Another subclass of operators $T,$ where $A$ or $B$ has a bounded inverse, leads to a similar analysis. This gives rise to a reversed version of Sebestyén inequality which is introduced in the present paper. It is shown that this second subclass, where $A^{-1}$ or $B^{-1}$ is bounded, can be characterized in a similar way by means of quasi-affinity of $T,$ rather that $T^*,$ to an operator $S=S^*\geq 0$. Furthermore, the connection between these two classes and weak-similarity as well as quasi-similarity to some $S=S^*\geq 0$ is investigated. Finally, the special case where $ S$ is bounded is considered. |
| title | Product of nonnegative selfadjoint operators in unbounded settings |
| topic | Functional Analysis 47A62, 47B02, 47B25, 47A06 |
| url | https://arxiv.org/abs/2507.14404 |