#### Title

SPECTRAL DUALITY FOR UNBOUNDED OPERATORS

#### 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

#### ISSN

0379-4024

#### Recommended Citation

"SPECTRAL DUALITY FOR UNBOUNDED OPERATORS" (2011). *Faculty Bibliography 2010s*. 1269.

http://stars.library.ucf.edu/facultybib2010/1269

## Comments

Authors: contact us about adding a copy of your work at STARS@ucf.edu