Geronimus spectral transforms and measures on the complex plane

We analyze a special spectral transform of a measure μ supported on a compact subset C of the complex plane. A perturbation μ1 of μ is said to be a Geronimus spectral transform if dμ1=dμ|z-α|2 where α∉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 μ1.