Unbounded Hermitian operators and relative reproducing kernel Hilbert space (Q607413)

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.