Journal of Operator Theory

Volume 65, Issue 2, Spring 2011 pp. 325-353.

Spectral duality for unbounded operators

Authors: Dorin Ervin Dutkay (1) and Palle E.T. Jorgensen (2)
Author institution: (1) University of Central Florida, Department of Mathematics, 4000 Central Florida Blvd. P.O. Box 161364, Orlando, FL 32816-1364 U.S.A.
(2) University of Iowa, Department of Mathematics, 14 MacLean Hall, Iowa City, IA 52242-1419, U.S.A.

Summary: 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

$(\Delta v)(x):=\ip{\delta_x}{v}_{\H}.$ Similarly, for every finite subset

$F\subset X$ , we get an operator

$\Delta_F$ . If

$F_1\subset F_2\subset\cdots$ is an ascending sequence of finite subsets such that

$\bigcup\limits_{k\in\bn}F_k=X$ , we are interested in the following two problems: (a) obtaining an approximation formula

$\lim\limits_{k\rightarrow\infty}\Delta_{F_k}=\Delta$ ; (b) establish a computational spectral analysis for the truncated operators

$\Delta_F$ in (a).

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