Title

SPECTRAL DUALITY FOR UNBOUNDED OPERATORS

Authors

Authors

D. E. Dutkay;P. E. T. Jorgensen

Comments

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Abbreviated Journal Title

J. Operat. Theor.

Keywords

Unbounded operators; reproducing kernel; Brownian motion; discrete; Laplacian; difference operators; Hermitian operator; extensions; spectral theory; KERNEL HILBERT-SPACES; HEISENBERG-GROUP; LAPLACIANS; EXTENSIONS; WAVELETS; FRACTALS; SYSTEMS; SERIES; Mathematics

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 delta(x), for x is an element of X, are contained in H. And we then define an associated operator Delta in H given by (Delta v)(z) := (H). Similarly, for every finite subset F subset of X, we get an operator Delta(F). If F-1 subset of F-2 subset of ... is an ascending sequence of finite subsets such that U F-k = X, we are interested in the following two problems: k is an element of N (a) obtaining an approximation formula lim k ->infinity Delta(Fk) = Delta; (b) establish a computational spectral analysis for the truncated operators Delta(F) in (a).

Journal Title

Journal of Operator Theory

Volume

65

Issue/Number

2

Publication Date

1-1-2011

Document Type

Article

Language

English

First Page

325

Last Page

353

WOS Identifier

WOS:000290824300006

ISSN

0379-4024

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