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arXiv · 2510.10935

Extensions of Operator-Valued Kernels on $\mathbb{F}^{+}_{d}$

James Tian
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Abstract

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 one-step dominance inequality on its interior, a global extension that preserves this interior data and the dominance property is always possible. For the problem of matching the kernel on the boundary, we introduce a 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.

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James Tian. 2025-10-13. Extensions of Operator-Valued Kernels on $\mathbb{F}^{+}_{d}$. https://arxiv.org/abs/2510.10935

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