Laurent polynomial perturbations of linear functionals. An inverse problem

Given a linear functional ℒ in the linear space ℙ of polynomials with complex coefficients, we analyze those linear functionals ℒ∼ such that, for a fixed α Ε ℂ, (ℒ∼, (z + z ^-1 - (α + ᾱ))p) = (ℒ, p) for every p Ε ℙ. We obtain the relation between the corresponding Carathéodory functions in such a way that a linear spectral transform appears. If ℒ is a positive definite linear functional, the necessary and sufficient conditions in order for ℒ∼ to be a quasi-definite linear functional are given. The relation between the corresponding sequences of monic orthogonal polynomials is presented.