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 〈· , ·〉 = 〈· , ·〉 H . We will be assuming that the Dirac masses δx, for x ∈ X, are contained in H. And we then define an associated operator ∆ in H given by (∆v)(x) := 〈δx , v〉H . Similarly, for every finite subset F ⊂ X, we get an operator ∆F . If F1 ⊂ F2 ⊂ . . . is an ascending sequence of finite subsets such that ∪k∈NFk = X, we are interested in the following two problems: (a) obtaining an approximation formula