2008/08/04 by Dorin Ervin Dutkay, Dutkay, Dorin Ervin, Palle E. T. Jørgensen +1
Mathematics · #18A30 #31C20 #34A45 #34L16 #37A50 #46E22 #47A75 #47B39 #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Mathematical Analysis and Transform Methods #Numerical Analysis (math.NA) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.0808.0485
openalex publication_date 2008/08/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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⋅⋅=\ip⋅⋅\H. We will be assuming that the Dirac masses δx, for x∈ X, are contained in \H. And we then define an associated operator Δ in \H given by (Δv)(x):=\ipδxv\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 ∪k∈\bnFk=X, we are interested in the following two problems: (a) obtaining an approximation formula limk→∞ΔFk=Δ; and (b) establish a computational spectral analysis for the truncated operators ΔF in (a).