Extensions of Operator-Valued Kernels on $\mathbb{F}^{+}_{d}$
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911206699171840 |
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| author | Tian, James |
| author_facet | Tian, James |
| contents | We study the problem of extending a positive-definite operator-valued kernel, defined on words of a fixed finite length from a free semigroup, to a global kernel defined on all words. We show that if the initial kernel satisfies a natural one-step dominance inequality on its interior, a global extension that preserves this interior data and the dominance property is always possible. This extension is constructed explicitly via a Cuntz-Toeplitz model. For the problem of matching the kernel on the boundary, we introduce an intrinsic shift-consistency condition. We prove this condition is sufficient to guarantee the existence of a global extension that agrees with the original kernel on its entire domain. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_10935 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Extensions of Operator-Valued Kernels on $\mathbb{F}^{+}_{d}$ Tian, James Functional Analysis Primary: 47A20, secondary: 44A60, 46E22, 47A57, 47B32 We study the problem of extending a positive-definite operator-valued kernel, defined on words of a fixed finite length from a free semigroup, to a global kernel defined on all words. We show that if the initial kernel satisfies a natural one-step dominance inequality on its interior, a global extension that preserves this interior data and the dominance property is always possible. This extension is constructed explicitly via a Cuntz-Toeplitz model. For the problem of matching the kernel on the boundary, we introduce an intrinsic shift-consistency condition. We prove this condition is sufficient to guarantee the existence of a global extension that agrees with the original kernel on its entire domain. |
| title | Extensions of Operator-Valued Kernels on $\mathbb{F}^{+}_{d}$ |
| topic | Functional Analysis Primary: 47A20, secondary: 44A60, 46E22, 47A57, 47B32 |
| url | https://arxiv.org/abs/2510.10935 |