Geronimus spectral transforms and measures on the complex plane

Abstract

We analyze a special spectral transform of a measure $\mu $ supported on a compact subset $C$ of the complex plane. A perturbation $\mu _{1}$ of $\mu $ is said to be a Geronimus spectral transform if d$\mu_1 = \frac {\text d \mu}{ -\alpha 2}$ where $\alpha \notin C$. We focus our attention in the analysis of the Hessenberg matrix associated with the multiplication operator in terms of the orthogonal polynomial basis defined by the measure $\mu _{1}$.