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 μ 1 of μ is said to be a Geronimus spectral transform if d μ 1 = d μ | 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 μ 1 .