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Scalar spectral measures associated with an Operator-Fractal | | Palle E. T. Jorgensen
; Keri A. Kornelson
; Karen L. Shuman
; | | Date: |
23 Apr 2012 | | Abstract: | We examine the operator $U_5$ defined on $L^2(mu_{frac14})$ where
$mu_{frac14}$ is the 1/4 Cantor measure. The operator $U_5$ scales the
elements of the canonical exponential spectrum for $L^2(mu_{frac14})$ by 5
--- that is, $Ue_{gamma} = e_{5gamma}$ where $e_{gamma}(t) = e^{2pi i
gamma t}$. It is known that $U_5$ has a self-similar structure, which makes
its spectrum, which is currently unknown, of particular interest. In order to
better understand the spectrum of $U_5$, we demonstrate a decomposition of the
projection valued measures and scalar spectral measures associated with $U_5$.
We are also able to compute associated Radon-Nikodym derivatives between the
scalar measures. Our decomposition utilizes a system of operators which form a
representation of the Cuntz algebra $mathcal{O}_2$. | | Source: | arXiv, 1204.5116 | | Services: | Forum | Review | PDF | Favorites | |
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