Spectral duality for a class of unbounded operators

Abstract

We establish a spectral duality for certain unbounded operators in Hilbert space. The class of operators includes discrete graph Laplacians arising from infinite weighted graphs. The problem in this context is to establish a practical approximation of infinite models with suitable sequences of finite models which in turn allow (relatively) easy computations. Let $X$ be an infinite set and let $\H$ be a Hilbert space of functions on $X$ with inner product $\ip{\cdot}{\cdot}=\ip{\cdot}{\cdot}_{\H}$. We will be assuming that the Dirac masses ${\delta }_x$, for $x\in X$, are contained in $\H$. And we then define an associated operator $\Delta$ in $\H$ given by $$(Δv)(x):=\ip{δ_x}{v}_{\H}.$$ Similarly, for every finite subset $F\subset X$, we get an operator ${\Delta }_F$. If $F_1\subset F_2\subset ...$ is an ascending sequence of finite subsets such that $\cup_{k\in\bn}F_k=X$, we are interested in the following two problems: (a) obtaining an approximation formula $$\underset{k\to \infty }{\lim }{\Delta }_{F_k}=\Delta ;$$ and (b) establish a computational spectral analysis for the truncated operators ${\Delta }_F$ in (a).

References in corpus (4)

• Bell Inequalities for Graph States
• Operator theory of electrical resistance networks
• Spectral Convergence of the Discrete Laplacian on Models of a Metrized Graph
• Spectral Theory for Discrete Lapacians