Inversion of perturbed linear operators that are singular at the origin

We consider the inversion of perturbed linear operators on Hubert space. Namely, we study linear operators that depend on a small parameter and are singular when the parameter is equal to zero. First we consider the affine dependence on the parameter. We treat subsequently the cases of bounded operators with closed range, bounded operators with non closed range, and densely defined and closed unbounded operators. Then, we extend the results to the cases of polynomial and analytic perturbations.