Scalar spectral measures associated with an Operator-Fractal

TL;DR

This paper investigates the spectral properties of a self-similar operator on a Cantor measure space, providing a decomposition of spectral measures and computing Radon-Nikodym derivatives using Cuntz algebra representations.

Contribution

It introduces a novel decomposition of spectral measures for a self-similar operator linked to the Cantor measure, advancing understanding of its spectral structure.

Findings

Decomposition of projection valued measures and scalar spectral measures.

Explicit computation of Radon-Nikodym derivatives between scalar measures.

Application of Cuntz algebra representations to spectral analysis.

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, $Ue_{\gamma} = e_{5\gamma}$ where $e_{\gamma}(t) = e^{2\pi i \gamma t}$. 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 $\mathcal{O}_2$.