SPECTRAL DUALITY FOR UNBOUNDED OPERATORS

    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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