When Does a Discrete Differential Perturbation of a Sequence of Orthonormal Polynomials Belong tol2?

Let μ be a positive determinate measure with moments of every order and infinite support and let (pn)nbe the corresponding sequence of orthonormal polynomials. Let (z1, k1), …, (zN, kN) be given, where thezʹs are different complex numbers and thekʹs are nonnegative integers. We characterize the operators of the formT=∑Nl=1 ∑klj=0 al, j δ(j)zl, for which the sequence (T(pn))nbelongs to ℓ2.