2012/04/23 by Jorgensen, Palle E. T., Kornelson, Keri A., Shuman, Karen L.
#FOS: Mathematics #Operator Algebras (math.OA) #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.1204.5116
We examine the operator U5 defined on L2(μ\frac14) where μ\frac14 is the 1/4 Cantor measure. The operator U5 scales the elements of the canonical exponential spectrum for L2(μ\frac14) by 5 --- that is, Ueγ = e5γ where eγ(t) = e2πi γt. It is known that U5 has a self-similar structure, which makes its spectrum, which is currently unknown, of particular interest. In order to better understand the spectrum of U5, we demonstrate a decomposition of the projection valued measures and scalar spectral measures associated with U5. 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 O2.