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Unbounded Hermitian operators and relative reproducing kernel Hilbert space. (English) Zbl 1220.47029
The paper is concerned with the theory of selfadjoint extensions of hermitian linear operators on a Hilbert space. The main application is the Laplace operator on a infinite weighted graph.
A duality pair $$(\Delta, \mathcal C)$$ consists of a densely defined hermitian operator $$\Delta$$ on a Hilbert space $$H$$ and a closed subspace $$\mathcal C$$ with $$\mathcal C \subseteq \ker(\Delta^*)$$. In this case, $$\Delta$$ admits a hermitian extension $$\Delta_{\mathcal C}$$ to $$\mathcal D(\Delta)+\mathcal C$$ and the domain of its adjoint is $$\mathcal D(\Delta_{\mathcal C}^*) = H \ominus \mathcal C$$.
The author investigates the reproducing kernel Hilbert space with base point $$o$$, $$(H,X,o)$$, under the additional assumptions that all Dirac functions $$\delta_x$$, $$x\in X$$, belong to $$H$$ and that $$H = \overline{\text{span}}\{k(\cdot, x) | x\in X, x\neq o\}$$ ($$k$$ is the kernel function of $$H$$). He shows that, under this condition, the operator $$\Delta|_V$$ defined by $$(\Delta|_V f)(x) = \langle \delta_x,\, f\rangle$$ with domain $$\text{span}\{ k_x - k_o| x\in X, x\neq o\}$$ is densely defined and positive. Moreover, $$(\Delta|_V, \mathcal C)$$ is a duality pair, where $$\mathcal C = \{ u \in H : \langle y,\, \delta_x \rangle = 0,\;x \in X\}$$, so $$\Delta|_V$$ can be extended to a hermitian operator $$\Delta_H$$ on $$\mathcal D(\Delta|_V) + \mathcal C$$. It is shown that $$\Delta_H$$ is essentially selfadjoint if and only if $$\Delta|_F$$ with domain $$\mathcal D(\Delta|_F) = \text{span}\{ \delta_x | x\in X\}$$ is essentially selfadjoint. The paper concludes with an application to an infinite weighted graph.

##### MSC:
 47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces) 46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) 47B25 Linear symmetric and selfadjoint operators (unbounded)
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