Hermitian indices and factorization of selfadjoint operators on a Kreĭn space
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918314472636416 |
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| author | Dritschel, Michael A. Maestripieri, Alejandra Rovnyak, James |
| author_facet | Dritschel, Michael A. Maestripieri, Alejandra Rovnyak, James |
| contents | The hermitian indices of a selfadjoint operator $C$ on a Kreĭn space $\mathcal H$ are defined as geometric measures of positivity and negativity of the operator. A different pair of indices arises in the Bognár-Krámli factorization of $C$, which writes $C$ as a product $AA^*$ where $A$ acts on a Kreĭn space $\mathcal A$ into $\mathcal H$ and has zero kernel; the new indices are the positive and negative indices of $\mathcal A$. Such factorizations are far from unique. When $\mathcal H$ is separable, it is known that the two notions of indices always coincide, and this has applications to index formulas in the theory of Julia operators and completion problems for operator matrices. A new proof of the equality of indices that does not require separability is given in this work. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_22366 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Hermitian indices and factorization of selfadjoint operators on a Kreĭn space Dritschel, Michael A. Maestripieri, Alejandra Rovnyak, James Functional Analysis 46C20, 46B50 The hermitian indices of a selfadjoint operator $C$ on a Kreĭn space $\mathcal H$ are defined as geometric measures of positivity and negativity of the operator. A different pair of indices arises in the Bognár-Krámli factorization of $C$, which writes $C$ as a product $AA^*$ where $A$ acts on a Kreĭn space $\mathcal A$ into $\mathcal H$ and has zero kernel; the new indices are the positive and negative indices of $\mathcal A$. Such factorizations are far from unique. When $\mathcal H$ is separable, it is known that the two notions of indices always coincide, and this has applications to index formulas in the theory of Julia operators and completion problems for operator matrices. A new proof of the equality of indices that does not require separability is given in this work. |
| title | Hermitian indices and factorization of selfadjoint operators on a Kreĭn space |
| topic | Functional Analysis 46C20, 46B50 |
| url | https://arxiv.org/abs/2601.22366 |