Scalar spectral measures associated with an Operator-Fractal

Abstract

We examine the operator $U_5$ defined on $L^2({\mu }_{\frac{1}{4}})$ where ${\mu }_{\frac{1}{4}}$ is the 1/4 Cantor measure. The operator $U_5$ scales the elements of the canonical exponential spectrum for $L^2({\mu }_{\frac{1}{4}})$ by 5 --- that is, $U{e}_{\gamma }=e_{5\gamma }$ 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 ${\mathcal{O}}_2$.