Title

Spectral Duality For Unbounded Operators

Keywords

Brownian motion; Difference operators; Discrete Laplacian; Extensions; Hermitian operator; Reproducing kernel; Spectral theory; 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 D in H given by (Δv) (x):= 〈 δx,v〉 H. Similarly, for every finite subset F ⊂ X, we get an operator ΔF. If F1 ⊂ F2 · ⊂ · · is an ascending sequence of finite subsets such that Uk∈N{double-struck} Fk=X, we are interested in the following two problems: (a) obtaining an approximation formula limk→∞ ΔFk = Δ; (b) establish a computational spectral analysis for the truncated operators ΔF in (a). © Copyright by Theta, 2011.

Publication Date

3-1-2011

Publication Title

Journal of Operator Theory

Volume

65

Issue

2

Number of Pages

325-353

Document Type

Article

Personal Identifier

scopus

Socpus ID

79955694719 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/79955694719

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